On split Malcev Poisson algebras

We introduce the class of split Malcev Poisson algebras as the natural extension of split (noncommutative) Poisson algebras. We show that if $P$ is a split Malcev Poisson algebra then $P = \oplus_{j \in J}I_j$ with $I_j$ a nonzero ideal of $P$ such that $\{I_{j_1},I_{j_2}\} = I_{j_1}I_{j_2}= 0$ for $j_1 \neq j_2$. Under some conditions, the above decomposition of $P$ involves a family of the simple ideals of $P$.