Let $\mathcal{A}$ be a nonassociative algebra such that $\mathcal{A}=\mathcal{A}_1\oplus\dots\oplus\mathcal{A}_n$ with $\mathcal{A}_i\subset\mathcal{A}$ an idempotent ideal.
I want to show that the derivation algebra $\mathcal{D}$ of $\mathcal{A}$ splits as the direct sum $\mathcal{D}=\mathcal{D}_1\oplus\dots\oplus \mathcal{D}_n$ where $\mathcal{D}_i$ is an ideal in $\mathcal{D}$ isomorphic to the derivation algebra of $\mathcal{A}_i$.
I'm getting stuck on what to show exactly, obviously it's enough to consider the case where $\mathcal{A}=\mathcal{A}_1\oplus\mathcal{A}_2$. Then we want to show $\mathcal{D}_1\cap\mathcal{D}_2=0$ and $\mathcal{D}_1+\mathcal{D}_2=\mathcal{D}$, and that $[\mathcal{D}_i,\mathcal{D}]\subset\mathcal{D}_i$ but I'm having trouble showing these directly.
This is proved for associative algebras by Jacobson in "Abstract derivation and Lie algebras", Trans. Amer. Math. Soc. 42 (1937), 206-224.