The Tits Construction and Some Simple Lie Superalgebras in Characteristic 3

Some simple Lie superalgebras, specific of characteristic 3, defined by S. Bouarroudj and D. Leites [BL06], will be related to the simple alternative and commutative superalgebras discovered by I.P. Shestakov [She97]. Throughout the paper, the ground field k will always be assumed to be of characteristic 6= 2. 1. Tits construction In 1966 [Tit66], Tits gave a unified construction of the exceptional simple classical Lie algebras by means of two ingredients: a unital composition algebra and a degree three simple Jordan algebra. The approach used by Benkart and Zelmanov in [BZ96] will be followed here (see also [EO]) to review this construction. Let C be a unital composition algebra over the ground field k with norm n. Thus, C is a finite dimensional unital k-algebra, with the nondegenerate quadratic form n : C → k such that n(ab) = n(a)n(b) for any a, b ∈ C. Then, each element satisfies the degree 2 equation a − t(a)a+ n(a)1 = 0, (1.1) where t(a) = n(a, 1) (