The splittest Kac superalgebra K10

We present an isotope of the usual version of the Kac 10-dimensional Jordan superalgebra K10 over a general ring of scalars Φ (isomorphic to the original version when i, , but not in characteristic 2), which we take as the “correct” split model for the simple superalgebra in all characteristics. This J=A⊕M has unit the sum of three reduced orthogonal idempotents. We exhibit a “quaternionic” model J⊆(H⊗H⊞Φf)⊕H of the bimodule structure for this model and the original one, as well as an “exterior” model J≅Λ2(M)⊕M for both the bimodule structure and the odd product. We give a reference table for all quadratic and triple products, and use this to explicitly describe all inner superderivations. In a subsequent article [K. McCrimmon, The Grassmann envelope of the Kac superalgebra K10, J. Algebra, in press. [5]] we will use this table to investigate the structure of the Grassmann envelope.