Let $\mathfrak{k}$ be a simple, compact (in the strict sense, i.e. the Killing form $B$ is negative definite) real Lie algebra.
Can $\mathfrak{k}$ have a direct sum decomposition of the form $\mathfrak{k}=\mathfrak{k}_1\oplus\mathfrak{k}_2$, where each $\mathfrak{k}_i$ is a subalgebra?
In principle I don't see why it shouldn't, but I can't come up with an example.