My question in the most simple form:

Let $\mathfrak{g}=\mathfrak{g}_1\oplus \mathfrak{g}_2$ be a direct sum of simple finite-dimensional Lie algebras over $\mathbb{C}$ and let $M$ be a finite-dimensional simple $\mathfrak{g}$-module (it is known that $M$ is some $V(\lambda)$ highest weight module).

Since $\mathfrak{g}_1$ is simple, $M$ decomposes into a direct sum of simple $\mathfrak{g}_1$-modules. What is that decomposition?

More generally, given an irreducible module over $\mathfrak{g}$ how does it decomposes over a subalgebra (simple, semisimple) of $\mathfrak{g}.$ Has it been investigated in some circumstances? I would be grateful for any links.