# commutator ideal for direct sum

Is commutator ideal compatible with direct sum?

Let's take $$\mathfrak{sl}_2(\Bbb K)\oplus\mathfrak{sl}_2(\Bbb K)$$ which is semi-simple because $$\mathfrak{sl}_2(\Bbb K)$$ is simple Lie algebra.

So we know that $$D(\mathfrak{sl}_2(\Bbb K))=\mathfrak{sl}_2(\Bbb K)\oplus \mathfrak{sl}_2(\Bbb K) = D(\mathfrak{sl}_2(\Bbb K))\oplus D(\mathfrak{sl}_2(\Bbb K))$$.

Is it true for any direct sum of simple Lie algebras?

Thank you for your hint or help.