An extended Freudenthal magic square in characteristic 3 (Q2466931)

The well-known Tits construction builds a Lie algebra from a unital composition algebra, a central simple Jordan algebra of degree three, and their Lie algebras of inner derivations. This construction is valid over arbitrary fields of characteristic not two or three and all the exceptional simple Lie algebras can be realized in this way. \textit{G. Benkart} and \textit{E. Zelmanov} [Invent. Math. 126, No. 1, 1--45 (1996; Zbl 0871.17024)] as well as \textit{G. Benkart} and the second author of the paper under review [Q. J. Math. 54, No. 2, 123--137 (2003; Zbl 1045.17002)] extended the Tits construction and the Freudenthal magic square to some of the exceptional classical Lie superalgebras. More recently, \textit{C. H. Barton} and \textit{A. Sudbery} [Adv. Math. 180, No. 2, 596--647 (2003; Zbl 1077.17011)] as well as \textit{J. M. Landsberg} and \textit{L. Manivel} [Adv. Math. 171, No. 1, 59--85 (2002; Zbl 1035.17016)] describe the exceptional simple Lie algebras as octonionic analogues of the classical matrix Lie algebras based on two composition algebras and their Lie algebras of triality. This leads to an alternative construction of the Freudenthal magic square and thereby explains its symmetry. Another advantage of this construction is that it remains valid over fields of characteristic three. Moreover, non-trivial composition superalgebras (i.e., with non-zero odd part) only exist in characteristic three and dimensions three or six. Using the latter in the construction of Barton and Sudbery yields an extended Freudenthal magic square in which Lie superalgebras appear. The aim of the paper under review is to give an explicit description of these Lie superalgebras which are either simple or contain a simple ideal of codimension one and with only a single exception have no counterpart in characteristic zero. Furthermore, all the split Lie superalgebras in the extended Freudenthal magic square are described as contragredient Lie superalgebras.