The Witt algebra (denoted by $W$) is an infinite dimensional Lie algebra as:

$[L_{m},L_{n}]=(m-n)L_{m+n}; \,\,\,\ m, n\in \mathbb{Z}$.

I am looking for second adjoint cohomology $H^{2}(W_{1}\oplus W_{2};W_{1}\oplus W_{2})$ and the question is: Is there any theorem about decomposition of this formula? For instance, is it possible to decompose this relation so that $$H^{2}(W_{1}\oplus W_{2};W_{1}\oplus W_{2})$$ $$=H^{2}(W_{1};W_{1})\oplus H^{2}(W_{1};W_{2})\oplus H^{2}(W_{2};W_{1})\oplus H^{2}(W_{2};W_{2})\;?$$

The commutation relations of $W_{1}\oplus W_{2}$ are

$$[L_{m},L_{n}]=(m-n)L_{m+n};$$

$$[L_{m},M_{n}]=0;$$

$$[M_{m},M{n}]=(m-n)M_{m+n}.$$