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.


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