A Kronecker factorization theorem for the exceptional Jordan superalgebra (Q1861470)

The exceptional Jordan superalgebra \(K_{10}\) of dimension 10 was introduced by \textit{V. Kac} in [Commun. Algebras 5, 1375--1400 (1977; Zbl 0367.17007)]. \(K_{10}\) corresponds to the exceptional 40-dimensional Lie superalgebra \(F(4)\). Suppose that a Jordan superalgebra \(\tilde J\) over a field of characteristic zero contains \(J=K_{10}\). Then \(\tilde J = (J\otimes S)\oplus J'\) where \(S\) is an associative commutative superalgebra. This main result can be applied to a classification of Lie superalgebras graded by the root system of \(F(4)\) obtained by \textit{S. Berman} and \textit{R. V. Moody} [Invent. Math. 108, 323--347 (1992; Zbl 0778.17018)].