From octonions to composition superalgebras via tensor categories

The nontrivial unital composition superalgebras, of dimension 3 and 6, which exist only in characteristic 3, are obtained from the split Cayley algebra and its order 3 automorphisms, by means of the process of semisimplification of the symmetric tensor category of representations of the cyclic group of order 3. Connections with the extended Freudenthal Magic Square in characteristic 3, that contains some exceptional Lie superalgebras specific of this characteristic are discussed too. In the process, precise recipes to go from (nonassociative) algebras in this tensor category to the corresponding superalgebras are given.


Introduction
In [Eld06], Lie algebras g over a field F that admit a Z/2-grading such that the even part is the direct sum of sl 2 (F) and another ideal d, and its odd part is, as a module for the even part, a tensor product of the two-dimensional natural module for sl 2 (F) and a module T for d, were considered.Thus, we have (1.1) In this case, T becomes a so-called symplectic triple system, and the invariance of the Lie bracket under the action of sl 2 (F) forces the bracket of odd elements to present the following form: for all u, v ∈ F 2 and x, y ∈ T , for a skew-symmetric bilinear form (• | •) on T and a symmetric bilinear map T × T → d, (x, y) → d x,y ; where u | v is, up to scalars, the unique sl 2 (F)-invariant bilinear form on F 2 , and All classical simple Lie algebras can be obtained in this way.But the main point raised in [Eld06] was that, in case the characteristic of F is 3, then the Z/2-graded vector space d ⊕ T , with bracket given by the bracket in d, the action of d in T , and by [x, y] = d x,y for x, y ∈ T , endows d ⊕ T with a structure of Lie superalgebra.This remark gave the construction of a family of new simple contragredient simple Lie superalgebras specific of characteristic 3. Another family of such simple Lie superalgebras was obtained in [Eld06] by means of simple orthogonal triple systems, and most of these new simple Lie superalgebras appeared in a unified way in the Extended Freudenthal Magic Square in [CE07].(See also [BGL09].) Quite recently [Kan22], Arun S. Kannan considered a much more general and surprising way of passing from Lie algebras to Lie superalgebras, obtaining the simple Lie superalgebras mentioned above in a quite combinatorial way.Another exceptional Lie superalgebra specific of characteristic 5, first obtained in [Eld07], is obtained too by Kannan, using a variation of his method in characteristic 5.
Kannan considered, over fields of characteristic 3, exceptional simple Lie algebras endowed with a nilpotent derivation d with d 3 = 0.This allows to view the Lie algebra as a Lie algebra in the category Rep α 3 of representations of the affine group scheme α 3 : R → {r ∈ R | r 3 = 0} (the kernel of the Frobenius endomorphism of the additive group scheme G a ).The semisimplification of Rep α 3 is the Verlinde category Ver 3 , which is equivalent to the category of vector superspaces.In this way, a path is obtained from Lie algebras in Rep α 3 to Lie superalgebras.
For Lie algebras as in (1.1), we may choose d to be the adjoint action by ( 0 1 0 0 ).In this case, the ideal sl 2 (F) constitutes a Jordan block of length 3 for d.The ideal d is annihilated by d, and the odd part F 2 ⊗ T is a direct sum of Jordan blocks of length 2, as d is nilpotent of order 2 on F 2 .The semisimplification process in [Kan22] returns precisely the Lie superalgebra d ⊕ T above.
In this paper, we want to concentrate on another feature in characteristic 3.Only over fields of this characteristic there are nontrivial composition superalgebras (see [ElOk02]).Our goal is to obtain the two unital composition superalgebras with nontrivial odd part: B(1, 2) and B(4, 2), from the split Cayley algebra by the process of semisimplification.It must be remarked that these composition superalgebras appeared for the first time in Shestakov's work on prime alternative superalgebras [She97].Actually, we will not semisimplify from Rep α 3 as in [Kan22], but from the category Rep C 3 of representations of the cyclic group of order 3 (or equivalenty, from the category of representations of the constant group scheme C 3 ).In other words, instead of considering algebras with a nilpotent derivation d with d 3 = 0, we consider algebras endowed with an automorphism of order 3.The semisimplification of Rep C 3 is again the Verlinde category Ver 3 .
The paper is organized as follows.Section 2 will review the needed results from the categories mentioned above.Our basic reference for monoidal and tensor categories will be [EGNO15].Concise recipes will be given to describe the superalgebra obtained from an algebra in Rep C 3 by semisimplification.Section 3 will be devoted to considering composition algebras in a symmetric tensor category, to reviewing the known results on order 3 automorphisms of the Cayley algebras over fields of charactetistic 3, and to using the recipes in the previous section in order to obtain B(1, 2) and B(4, 2) from the split Cayley algebra.Section 4 will be devoted to showing how this process of semisimplification behaves with respect to algebras of derivations or of skew transformations relative to a nondegenerate symmetric bilinear form.Also, the Extended Freudenthal Magic Square in [CE07] is built in terms of composition superalgebras, and it will be shown in the last section how the work in Section 3 can be used to obtain the Lie superalgebras in the extended square by semisimplification from the algebras in the last row of the classical Freudenthal Magic Square, in a way different from the one considered in [Kan22].That is, semisimplification provides a bridge between the classical Freudenthal Magic Square and its extended version.
Throughout the paper, F will denote a ground field.All vector spaces will be assumed to be finite-dimensional over F and unadorned tensor products will be over F. Most of the time, the characteristic of F will be 3.

From algebras to superalgebras
This section will review, in a way suitable for our purposes, known results on the categories Rep C 3 , Ver 3 , and sVec.For details, the reader may consult [EtKa21,Ost15,EtOs19] and references there in.
Throughout this section, the characteristic of the ground field F will always be 3.
2.1.Semisimplification of Rep C 3 .The category Rep C 3 , whose objects are the finite-dimensional representations of the finite group C 3 over F or, equivalently, of the corresponding constant group scheme, and whose morphisms are the equivariant homomorphisms, is a symmetric tensor category, with the usual tensor product of vector spaces and the braiding given by the usual swap: Fix a generator σ of C 3 .Rep C 3 is not a semisimple category.The indecomposable objects are, up to isomorphism, where the action of σ is trivial on V 0 , and

nonuniquely, as
where For instance, End Rep C 3 (V 1 ) consists of those endomorphisms of V 1 which commute with σ.Any such endomorphism f satisfies f (v 0 ) = αv 0 +βv 1 and f (v 1 ) = αv 1 for scalars α, β ∈ F, so that f = αid V1 + g for a nilpotent endomorphism g.It fol- ) is again of the form αid V2 + g for a nilpotent endomorphism, but now tr(id V2 ) = 0 as the characteristic of F is 3, and it turns out that End Rep C 3 (V 2 ) consists entirely of negligible endomorphisms.
Negligible homomorphisms form a tensor ideal and this allows us to define the semisimplification of Rep C 3 , which is the Verlinde category Ver 3 , whose objects are the objects of Rep C 3 , but whose morphisms are given by This is again a symmetric tensor category, with the tensor product in Rep C 3 , and the braiding induced by the one in Rep C 3 .
Denote by where id X denotes the identity morphism in Rep C 3 (the identity map).We have thus obtained the semisimplification functor: (2.2) The semisimplification functor S is F-linear and braided monoidal (see [EGNO15, Definitions 1.2.3 and 8.1.7]).Some straightforward consequences of the definitions are recalled here: Properties 2.1.
• Ver 3 is semisimple: any object is isomorphic to a direct sum of copies of V 0 and V 1 .
is the direct sum of its submodule of symmetric tensors, which is isomorphic to V 2 , and its one-dimensional submodule of skew-symmetric tensors, which is isomorphic to V 0 .An explicit isomorphism (2.3) • The braiding in Ver 3 , for objects X, Y , is given by [c X,Y ], where c X,Y is the braiding in Rep C 3 (i.e., the swap x ⊗ y → y ⊗ x).Then, identifying 2.2.Equivalence of Ver 3 and sVec.This equivalence is well known, but concrete formulas for these equivalence will be needed later on, and hence this will be reviewed in some detail.The objects of the category sVec of vector superspaces (over our ground field F) are the Z/2-graded vector spaces X = X0 ⊕ X1, and the morphisms f : X → Y are the linear maps preserving this grading: f (X0) ⊆ Y0, f (X1) ⊆ Y1.We will write f = f0 ⊕ f1, with f ā : X ā → Y ā given by the restriction of f , a = 0, 1.This is a symmetric tensor category, with the braiding given by the parity swap: for homogeneous elements x, y, where (−1) xy is −1 if both x and y are odd, and it is 1 otherwise.
The F-linear functor given on objects and morphisms by (2.4) is an equivalence of categories.Here the action of F is a monoidal functor with natural isomorphism J : , where j ′ X,Y is defined as follows: (2.6) Note that F preserves the braiding too.In other words, the following diagram is commutative for all X, Y : Therefore, the functor F is a braided monoidal equivalence.
2.3.Recipe to get superalgebras from algebras in Rep C 3 .Given a linear map m : A ⊗ B → C in sVec, the composition A) is an algebra in Ver 3 with multiplication given by the composition Now, given a homomorphism µ : A ⊗ B → C in Rep C 3 , our goal is to find explicitly objects A, B, C in sVec and a homomorphism m : A ⊗ B → C such that there are isomorphisms [ι A ] : that make the following diagram commutative: (2.7) In particular, given an algebra A in Rep C 3 , with multiplication µ(x ⊗ y) = xy, our goal is to find explicitly the superalgebra (A, m), unique up to isomorphism, such that the algebras F (A), F (m) • J A,A and (A, [µ]) are isomorphic algebras in Ver 3 .This is achieved in Corollary 2.8.
To begin with, note that the objects A, B, C in Rep C 3 decompose as in (2.1): (The projections are relative to the splitting In particular, if (A, µ) is an algebra in Rep C 3 (this means that σ acts as an algebra automorphism), then the previous lemma restricts as follows: Corollary 2.3.Let (A, µ) be an algebra in Rep C 3 , with µ(x ⊗ y) = xy for all x, y ∈ A. Pick a splitting as in (2.1).Then the algebra (A, [µ]) in Ver 3 is isomorphic to the algebra (The projections are relative to the splitting defined as in Lemma 2.2.Then, for any x ∈ A 1 and y ∈ B 1 , the following equation holds: and (2.8) follows.
Let µ : A⊗B → C be a homomorphism in Rep C 3 as in Remark 2.4.Fix splittings of A, B, C as in (2.1), and pick subspaces (2.9) and similarly for A and B. Consider the objects A = A0 ⊕ A1, B = B0 ⊕ B1, and Recipe 2.5.Take projections relative to the splitting (2.9), and define the homomorphism m : A ⊗ B → C in sVec as follows: The homomorphism m is a morphism in the category sVec.
Given any object A in Rep C 3 , take a splitting , and a refinement A = A0 ⊕A1 ⊕δ(A1)⊕A 2 as in (2.9).Consider the object A = A0 ⊕A1 in sVec, and the linear map ι A : F (A) → A defined as follows: (2.10) Proof.Because of Lemma 2.2, it is enough to prove that the diagram (in Ver 3 ) (2.11) is commutative.(Here we use the same notation ι A to denote the isomorphism Using the inverse of J A,B (see (2.6)), this is equivalent to checking that in the next diagram in Rep C 3 : so that Φ is trivial on A0 ⊗ B0.In the same vein, for x1 ∈ A1, y1 ∈ B1 we get and Φ is trivial too on A1 ⊗ B1.Now, for x0 ∈ A0 and y1 ∈ B1, and for α, β ∈ F, we get It follows that the restriction Φ| (A0⊗B1)⊗V1 , takes (A0 ⊗ B1) ⊗ V 1 , which is a direct sum of copies of V 1 , to δ(C1), which is a direct sum of copies of V 0 , and hence it is negligible.In the same vein, the restriction Φ| (A1⊗B0)⊗V1 is negligible.We conclude that Φ is negligible, as required.
The algebra (A, m) is an algebra in sVec (a superalgebra).
In this case, Theorem 2.6 restricts to the following Corollary: In other words, (A, m) is the superalgebra that corresponds to the 'semisimplification' of (A, µ).

From octonions to composition superalgebras
The notion of composition algebra on a symmetric tensor category over a field of characteristic not 2 will be considered here.The order 3 automorphisms of the Cayley algebras, i.e., of the eight-dimensional unital composition algebras, were determined in [Eld18].In particular, any such automorphism on a Cayley algebra over a field of characteristic 3 allows us to view the Cayley algebra as an algebra in Rep C 3 , and hence to obtain, through the semisimplification functor in (2.2), an algebra in Ver 3 and thus, through the equivalence F in (2.4), a composition superalgebra.
3.1.Composition algebras in a symmetric tensor category.A composition algebra over a field F is a triple (C, µ, n), where µ : C ⊗ C → C, µ(x ⊗ y) = xy is the multiplication of C, and n : C → F is a nonsingular multiplicative quadratic form, called the norm.Here nonsingular means that either the polar form n(x, y) := n(x + y) − n(x) − n(y) is a nondegenerate bilinear form, or the characteristic of F is 2 and there is no nonzero element such that n(x, C) = 0 = n(x).Note that the same symbol is used to denote the norm and its polar form.Also, the polar form may be considered as a linear map n : C ⊗ C → F. The norm being multiplicative means that the equation n(xy) = n(x)n(y) holds for all x, y ∈ C.
Unital composition algebras (also termed Hurwitz algebras) over a field are the analogues of the classical algebras or real and complex numbers, quaternions, and octonions.In particular their dimension is restricted to 1, 2, 4 or 8.The reader may consult [ZSSS82, Chapter 2], [KMRT98, Chapter VIII], or the survey paper [Eld21].
Assume in the rest of the section that the characteristic of the ground field F is not 2.
Linearizing twice the multiplicative identity one gets for all x, y, z, t ∈ C, and conversely, the characteristic being not 2, (3.1) gives, with z = x and t = y, the multiplicative condition n(xy) = n(x)n(y).Now, we may define a composition algebra in a symmetric tensor category C as an object A endowed with morphisms µ : A ⊗ A → A and n : A ⊗ A → 1, such that the following conditions are satisfied: The following equality of morphisms A ⊗4 → 1, generalizing (3.1), holds: where we omit the isomorphism 1⊗1 ≃ 1, and where is an isomorphism.(We omit the associative and unitor morphisms, and coev A denotes the coevaluation morphism 1 → A ⊗ A * in the symmetric tensor category C.) Assume now that the characteristic of the ground field F is 3, and let (A, µ, n) be a composition algebra endowed with an automorphism σ with σ 3 = id.(This means that σ leaves invariant both µ and n.)Then, looking at the polar form as a linear map n : A ⊗ A → F, the triple (A, µ, n) is a composition algebra in Rep C 3 .Lemma 3.1.Let (A, µ, n) be a composition algebra endowed with an automorphism σ with σ 3 = id.Let A = A 0 ⊕ A 1 ⊕ A 2 be a splitting as in (2.1).Then, with δ = σ − id, the following conditions hold: (1) n ker δ, δ(A) = 0, (2) n δ(A 1 ), δ(A) = 0, (3) n(δ(x), y) + n(x, δ(y)) = 0 for all x ∈ A 1 and y ∈ A.
Apply the semisimplification functor S in (2.2) to get a composition algebra (A, [µ], [n]) in Ver 3 . As and n ′ is given by the formula Recipe 2.5 with A = B and C = F gives the following: Recipe 3.3.Let (A, µ, n) be a composition algebra in Rep C 3 .Take A0 = A 0 and A1 as in (2.9), and define a bilinear map n on A = A0 ⊕ A1 (or equivalently a linear map A ⊗ A → F) as follows: In other words, (A, m, n) is the composition superalgebra that corresponds to the 'semisimplification' of (A, µ, n).

Order 3 automorphisms of Cayley algebras.
A unital composition algebra (or Hurwitz algebra) of dimension ≥ 2 is said to be split if its norm is isotropic.For each dimension 2, 4 or 8, there is a unique split Hurwitz algebra, up to isomorphism.The split Cayley algebra has a canonical basis with multiplication given in Table 1.The elements of the canonical basis are all isotropic and they form a hyperbolic basis: All the other values of the polar form for basic elements are either 0 or follow from the above by using that n is symmetric.Note that the u i 's generate the whole algebra.
Table 1.Multiplication table of the split Cayley algebra.
The subalgebra spanned by the orthogonal idempotents e 1 and e 2 is the split Hurwitz algebra in dimension 2, while the subalgebra spanned by e 1 , e 2 , u 1 , v 1 is the split quaternion algebra.
Among Cayley algebras (i.e., eight-dimensional Hurwitz algebras) over a field F of characteristic 3, only the split one is endowed with order 3 automorphisms.The order 3 automorphisms are then classified, up to conjugacy, in this theorem.
Theorem 3.5 ([Eld18, Theorem 6.3]).Let (C, µ, n) be a Cayley algebra over a field F of characteristic 3, and let σ be an order 3 automorphism of (C, µ, n).Then (C, µ, n) is the split Cayley algebra and one of the following conditions holds: (1) (σ − id) 2 = 0, and there exists a canonical basis of C such that (2) (σ − id) 2 = 0 and there is a quadratic étale subalgebra K of C fixed elementwise by σ.
If F is algebraically closed, then there is a canonical basis of C such that σ(u i ) = u i+1 (indices modulo 3).
(3) There is a canonical basis such that (4) There is a canonical basis such that It must be remarked that the automorphism in item (1) above corresponds to the so called quaternionic idempotents of Okubo algebras, while the automorphism in item (4) corresponds to the singular idempotents of Okubo algebras.These are specific to characteristic 3 and have no counterpart in other characteristics.For details, the reader may consult [Eld18].
3.3.'Semisimplification' of Cayley algebras.Assume in this section that the characteristic of the ground field F is 3.
For each of the possibilities in Theorem 3.5, the unital composition superalgebra that corresponds to the semisimplification of the Hurwitz algebra (C, µ, n) will be determined here.In order to do this, it is enough to apply Recipes 2.7 and 3.3.
(1) σ(u i ) = u i , i = 1, 2, σ(u 3 ) = u 3 + u 2 .Then e 1 , e 2 , v 1 , and v 3 are fixed by σ, while σ( The norm n restricts to n on the even part C0, and satisfies n( (2) There is a quadratic étale subalgebra K of C fixed elementwise by σ, and the action of σ on K ⊥ (orthogonal relative to n) is given by two cycles of length 3.This gives the decomposition in (2.1) with C 0 = K, C 1 = 0 and C 2 = K ⊥ .Then the semisimplification simply gives the composition algebra K with trivial odd component.

Semisimplification: skew transformations, derivations
This last section will show some features of the semisimplification process.The Lie algebra of derivations of an algebra (A, µ) in Rep C 3 is also an algebra in Rep C 3 in a natural way, but its semisimplification may fail to be the Lie superalgebra of derivations of the semisimplification of (A, µ).However, the semisimplification of the Lie algebra of the skew-symmetric transformations, relative to the norm, of a composition algebra in Rep C 3 is isomorphic to the orthosymplectic Lie superalgebra of skew-transformations (in the super setting) of the corresponding composition superalgebra.
Throughout the section, the characteristic of the ground field F will be assumed to be 3. 4.1.Skew transformations.Given an object V in Rep C 3 , we will denote by V ss an object in sVec, such that F (V ss ) and V (= S(V) for S in (2.2)) are isomorphic as objects in Ver 3 .The vector superspace V ss will be called a semisimplification of V.In the same vein, given an algebra (A, µ) in Rep C 3 , we will denote by (A ss , µ ss ) a superalgebra (i.e., an algebra in sVec) such that F (A ss ), F (µ ss ) • J A ss ,A ss is isomorphic to the algebra (A, [µ]) in Ver 3 .The multiplication µ will be omitted if it is clear from the context, The same applies to vector spaces endowed with a bilinear form: (V ss , b ss ); or to composition algebras (C ss , µ ss , n ss ).
Proposition 4.1.Let V be an object in Rep C 3 .
• The associative superalgebras End F (V ss ) and End F (V) ss are isomorphic.
• Let b : V ⊗ V → F be a morphism in Rep C 3 such that the bilinear form (denoted by the same symbol) given by b(x, y) := b(x ⊗ y) is symmetric and nondegenerate.Then the bilinear form corresponding to the morphism b ss : V ss ⊗ V ss → F in sVec is super-symmetric and nondegenerate, and the orthosymplectic Lie algebra osp(V ss , b ss ) is isomorphic to the semisimplification so(V, b) ss .
Proof.Note first that End F (V) is isomorphic to V ⊗ V * as objects in Rep C 3 , where the element v ⊗ f corresponds to the endomorphism w → vf (w), for v, w ∈ V and f ∈ V * .The multiplication in V ⊗ V * is given by the following composition (associative and unitor morphisms are omitted, as usual) involving the evaluation morphism ev V : The first part follows at once because the semisimplification functor S in (2.2) is a braided monoidal functor (see [EGNO15, Definition 8.1.7]),and the equivalence F in (2.4) is a braided monoidal equivalence.
For the second part, the symmetry of b ss in sVec (that is, the fact that b ss is super-symmetric) and its nondegeneracy are again consequences of the fact that S and F are braided monoidal functors.In this case, the algebra End F (V) is isomorphic to V ⊗ V, where the element v ⊗ w corresponds to the linear map x → vb(w, x), and the multiplication is given by the composition The corresponding orthogonal Lie algebra so(V, b) corresponds to the subspace Skew 2 (V ⊗ V) of skew-symmetric tensors, which is the image of the projection is the braiding (the usual swap in this case).
Since S and F are braided monoidal functors, the semisimplification so(V, b) ss is isomorphic to the image of the projection 1 2 (id V ss ⊗V ss − c ss V ss ,V ss ), where now the braiding c ss V ss ,V ss is given by the parity swap.This is the subspace of super-skewsymmetric tensors in V ss ⊗ V ss , and this, in turn, is isomorphic to the orthosymplectic Lie superalgebra osp(V ss , b ss ).4.2.Derivations.As mentioned at the beginning of the section, derivations present a different behavior under semisimplification.Note that any automorphism τ of an algebra (A, µ) induces an automorphism Ad τ : d → τ • d • τ −1 in its Lie algebra of derivations.
To begin with, given a Lie algebra (L, µ L ), an algebra (A, µ A ) in Rep C 3 , and a morphism Φ : L ⊗ A → A in Rep C 3 given by x ⊗ a → x.a; Φ is an action by derivations of L on A if and only if the following two conditions are satisfied for all x, y ∈ L and a, b ∈ A: where [x, y] = µ L (x ⊗ y) and ab = µ A (a ⊗ b).This can be written as follows: where the first equality holds in Hom Rep C 3 (L ⊗ L ⊗ A, A), while the second holds in Hom Rep C 3 (L ⊗ A ⊗ A, A), and therefore all this goes smoothly under semisimplification.
As a consequence, we obtain our next result: Proposition 4.2.For any algebra (A, µ) in Rep C 3 , there is a natural homomorphism Der(A, µ) ss → Der(A ss , µ ss ) from the semisimplification of the Lie algebra of derivations of (A, µ) into the Lie superalgebra of derivations of the superalgebra (A ss , µ ss ).
We will compute next the semisimplification of the Lie algebras of derivations of the algebras (C, µ, n) in cases (1), (3) and (4) of subsection 3.3.As in subsection 3.3, the situation in case (2) is quite trivial.
Take the canonical basis of the split Cayley algebra C as in Table 1, and write The characteristic of F being 3 implies that the Lie algebra of derivations Der(C) splits as (see [ElKo13,Proposition 4.29]) where, as usual, ad x (y) = [x, y], and where S = {d ∈ Der(C) | d(e 1 ) = 0 = d(e 2 )}.
At this point, it should be remarked that, the characteristic being 3, Der(C) is not the contragredient Lie algebra attached to the Cartan matrix 2 −3 −1 2 .(See [Kan22, Example 3.4].)This contragredient Lie algebra is, in fact, a subalgebra of Der(C) given by S ′ ⊕ ad U ⊕ ad V , with S ′ a subalgebra of S isomorphic to gl 2 (F).
Moreover, the restriction of S to U gives an isomorphism S ≃ sl(U).The subspace ad C is a seven-dimensional ideal of Der(C) isomorphic to the projective special linear Lie algebra psl 3 (F), and the quotient Der(C)/ad C is again isomorphic to psl 3 (F).Under the isomorphism S ≃ sl(U), any trace zero endomorphism f of U acts trivially on Fe 1 + Fe 2 , as f on U, and as −f * on V, where f * is determined by the equation n f (u), v = n u, f * (v) for all u ∈ U and v ∈ V. We will identify S with sl(U) and will denote by E ij the linear endomorphism of U taking u j to u i and sending u l to 0 for l = j.In particular, ad e1−e2 is identified with twice the identity map • In case (1) of subsection 3.3, and because Ad σ (ad x ) = ad σ(x) for all x, it is easy to compute a splitting of Der(C) into Jordan blocks relative to the nilpotent transformation ∆ := Ad σ − id, as follows: Therefore we get a splitting Der(C) = D0 ⊕ D1 ⊕ ∆(D1) ⊕ D 2 as in (2.9) with D0 = ad F(e1−e2)+Fu1+Fv1 and D1 = Fad u3 ⊕ Fad v2 ⊕ FE 12 ⊕ FE 31 .Recipe 2.7 gives the multiplication in the Lie superalgebra Der(C) ss = D0 ⊕ D1.The subspace D0 ⊕ Fad u3 ⊕ Fad v2 is an ideal isomorphic to osp(1, 2).
Moreover, the action of Der(C) ss on C ss = C0 ⊕ C1 is determined by its action on the odd part (as the odd part generates the whole superalgebra, see (3.3)).Using Recipe 2.5 we obtain It turns out that Der(C) ss = D0 ⊕ D1 is isomorphic to the Lie superalgebra Der(C ss ) (see [ElOk02,Theorem 5.8]).are all fixed by Ad σ , while the remaining three elements form a threedimensional indecomposable module for the action of Ad σ (i.e., isomorphic to V 2 in Rep C 3 ).The odd part decomposes into the direct sum of the following Jordan blocks for the linear endomorphism ∆ = Ad σ − id of Der(C): Therefore we get a splitting Der(C) = D0 ⊕ D1 ⊕ ∆(D1) ⊕ D 2 as in (2.9), with D1 = 0 as there are no Jordan blocks of length 2, and D0 = span {E 12 , E 21 , E 11 − E 22 , E 13 , E 23 }.It turns out that Der(C) ss is a Lie algebra (its odd part is trivial) of dimension 5, which is the direct sum of a copy of sl 2 (F) and a two-dimensional abelian ideal: FE 13 + FE 23 .This ideal is the natural two-dimensional module for the copy of sl 2 (F).By [ElOk02, proof of Lemma 5.3], Der(C ss ) is isomorphic to sl 2 (F).Actually, it turns out that the ideal FE 13 + FE 23 in Der(C) ss acts trivially on C ss .(Recall that the action is given by Recipe 2.5.)In this case, the natural homomorphism in sVec from Der(C) ss into Der(C ss ) is surjective.
• In case (4) of subsection 3.3, lengthy but straightforward computations give the following Jordan blocks for the action of the nilpotent endomorphism ∆ = Ad σ − id: In this case, the kernel of the natural homomorphism in sVec from Der(C) ss into Der(C ss ) is D1, and this homomorphism is neither injective nor surjective.

The extended Freudenthal Magic Square
Assume for a while that the characteristic of the ground field F is just different from 2.
Different authors [Vin66, BS03, LM02] have considered several symmetric constructions of Freudenthal's Magic square in terms of two unital composition algebras.We will follow here [Eld04], but restricted, for simplicity, to the use of the so-called para-Hurwitz algebras.Let (C, µ, n) and (C ′ , µ ′ , n ′ ) be two unital composition algebras over a field F of characteristic not 2. Denote in both cases the multiplication by juxtaposition, and consider the new multiplications μ given by μ(x, y) = x • y := x y, where x = n(x, 1) − x is the standard conjugation.Define similarly μ′ .Consider the associated triality Lie algebras: For simplicity, we will just write tri(C) and tri(C ′ ).
The vector space where ι i (C ⊗ C ′ ) is just a copy of C ⊗ C ′ (i = 0, 1, 2) becomes a Lie algebra with the bracket defined as follows: • the Lie bracket in tri(C) ⊕ tri(C ′ ), which thus becomes a Lie subalgebra of g, x, y ∈ C, and x ′ , y ′ ∈ C ′ , where t x,y is the element of tri(C) defined as follows: with s x,y : z → n(x, z)y − n(y, z)x, l x : z → x • z, and r x : z → z • x, and similarly for C ′ .The Lie algebras thus obtained are semisimple (simple in most cases) and, if the characteristic of the ground field F is neither 2 nor 3 then the type of the Lie algebras obtained is given by Freudenthal Magic Square, where the index over each row (respectively column) is the dimension of C (resp.C ′ ): If the characteristic of the ground field F is 3, instead of simple Lie algebras of type A 2 or A 5 we obtain forms of the projective general Lie algebra pgl 3 (F) or pgl 6 (F), and instead of simple Lie algebras of type E 6 , we obtain Lie algebras of dimension 78 whose derived ideal is simple of type E 6 (the simple Lie algebra of type E 6 has dimension 77 in characteristic 3).If (C, µ, n) is a Cayley algebra, then the projection π 0 : (d 0 , d 1 , d 2 ) → d 0 , gives a Lie algebra isomorphism tri(C, •, n) ≃ so(C, n).In other words, for any d 0 ∈ so(C, n) there are unique Hence the triples t x,y in (5.2) span the triality Lie algebra tri(C, •, n).
Therefore, the linear map ϑ : so(C, n) → so(C, n), given by is a Lie algebra automorphism that makes the following diagram commutative (θ as in (5.1)): (5.5) The natural and the two half-spin actions of so(C, n) are involved in the Lie bracket of g(C, C ′ ).The natural action Φ 0 is given by the composition where so(C, n) is identified with Skew 2 (C, n) as in Section 4.1.This composition behaves as follows: The two half-spin representations Φ 1 and Φ 2 are respectively the compositions: given by given by The commutativity of (5.4) is then equivalent to the commutativity of the following diagram:

Φi
(5.6) for i = 1, 2. Note that all the homomorphisms above are given in terms of the norm n, the multiplication μ and the braiding (the 'swap').This symmetric construction of Freudenthal's Magic Square was extended, over fields of characteristic 3, by using the unital composition superalgebras B(4, 2) and B(1, 2) in [CE07], thus obtaining an extended Freudenthal's Magic Square that includes many of the exceptional contragredient simple Lie superalgebras in characteristic 3.As before, in the second row or column, the superalgebras obtained are no longer simple, but their derived subalgebras are simple.
All these Lie superalgebras have been obtained by Kannan [Kan22] by considering nilpotent derivations of degree 3 of some of the simple exceptional Lie algebras, and thus looking at these as Lie algebras in the category Rep α 3 , whose semisimplification is again Ver 3 .
Actually, the semisimplification of Cayley algebras in Section 3 provides a bridge between the symmetric construction of Freudenthal's Magic Square and the extended square in [CE07].
Assume from now on that the characteristic of our ground field is 3. Any order 3 automorphism of a unital composition algebra (C, µ, n) is also an automorphism of its para-Hurwitz counterpart, and then it induces an order 3 automorphism of so(C, n) and of tri(C, •, n) commuting with the triality automorphism.
Therefore, starting with an order 3 automorphism σ of a Cayley algebra (C, µ, n) such that its semisimplification is isomorphic to either B(1, 2) or B(4, 2), there is an order 3 automorphism induced in g(C, C ′ ), where we combine the order 3 automorphism on C and the identity automorphism in C ′ .The action of this order 3 automorphism is as follows: As any automorphism of (C, µ, n) commutes with the standard conjugation x → x = n(1, x) − x, it turns out that the semisimplification of (C, μ, n) is the para-Hurwitz superalgebra (C ss , µ ss , n ss ).For these superalgebras, the projection π 0 : (d 0 , d 1 , d 2 ) → d 0 is a Lie superalgebra isomorphism (see [ElOk02]).Using Proposition 4.1, we get a chain of isomorphisms of Lie superalgebras: tri(C, μ, n) ss ≃ so(C, n) ss ≃ osp(C ss , n ss ) ≃ tri(C ss , µ ss , n ss ).
The commutativity of (5.4) shows that, under these isomorphisms, the Lie superalgebra automorphism θ ss of tri(C ss , µ ss , n ss ) corresponds to the automorphism ϑ ss of osp(C ss , n ss ), and the commutativity of (5.6), together with the fact that the Φ i 's are defined in terms of n, µ, and the braiding, shows that ϑ ss satisfies the 'super' versions of (5.3) and (5.5).But tri(C ss , µ ss , n ss ) is spanned by the 'super' versions of the triples t x,y in (5.2) (see [ElOk02]).It follows that, under the isomorphisms above, the automorphism θ ss of tri(C, μ, n) ss corresponds to the cyclic permutation (d 0 , d 1 , d 2 ) → (d 2 , d 0 , d 1 ) in tri(C ss , µ ss , n ss ), and, as a consequence, that the Lie superalgebra g(C, C ′ ) ss is isomorphic to the superalgebra g(C ss , C ′ ) in [CE07].
The same arguments work if both C and C ′ are Cayley algebras endowed with order 3 automorphisms.We also get an induced order 3 automorphism of g(C, C ′ ).These order 3 automorphisms allow us to see g(C, C ′ ) as a Lie algebra in Rep C 3 and get its semisimplification g(C, C ′ ) ss , which is isomorphic to g(C ss , C ′ss ).
In other words, the Lie superalgebras in the extended Freudenthal's Magic square can be obtained by semisimplification of the Lie algebras (in Rep C 3 ) in the fourth row of the classical Freudenthal's Magic Square in characteristic 3.
It must be pointed out here that in [Kan22], g B(1, 2), B(1, 2) is obtained from the exceptional Lie algebra of type E 6 endowed with a suitable nilpotent derivation of order 3, while the above comments show that g B(1, 2), B(1, 2) is obtained too from the exceptional Lie algebra of type E 8 , that is, from the Lie algebra g(C, C ′ ) where both C and C ′ are the split Cayley algebras, endowed with automorphisms of types (3) or (4) in Theorem 3.5.
Then Properties 2.1 immediately imply the following result: Lemma 2.2.Given objects A, B, C in Rep C 3 with the above decompositions, let µ : A ⊗ B → C be a homomorphism in Rep C 3 .Then the inclusion maps A ′ ֒→ A, B ′ ֒→ B, and C ′ ֒→ C, induce isomorphisms in Ver 3 , and the diagram turns out to be an isomorphism in Ver 3 .Theorem 2.6.Let µ : A ⊗ B → C be a homomorphism in Rep C 3 .Pick splittings of A, B, and C as in (2.1), and refinements as in (2.9).Define a homomorphism m : A ⊗ B → C in sVec by means of Recipe 2.5.Then, with ι A , ι B , ι C as in (2.10), the diagram (2.7) is commutative.
have ϑ 2 (s x,y ) = 1 2 r y l x − r x l y .
where Q is the quaternion subalgebra spanned by e 1 , e 2 , u 3 , v 3 and Q ⊥ (orthogonal subspace relative to the norm) is spanned byu 1 , u 2 , v 1 , v 2 .This is a Z/2-grading of C that induces a Z/2-grading of Der(C) whose even component is {d ∈ Der(C) | d(Q) ⊂ Q}is the span (with the notation used in the previous case) of E 12 , E 21 , E 11 −E 22 , ad e1−e2 , ad u3 , ad v3 .The derivations E 12 , E 21 , E 11 −E 22