Is there any Kunneth formula for the Lie algebra cohomology with coefficients in the adjoint representation

Let $$\mathfrak{g}$$ be a finite dimensional complex Lie algebra and let $$H^k(\mathfrak{g},\mathfrak{g})$$ denote the $$k$$-th cohomology group of $$\mathfrak{g}$$ with coefficients in the adjoint representation.

Let $$\mathfrak{h}$$ and $$\mathfrak{a}$$ be two finite dimensional complex Lie algebras, and let $$\mathfrak{g} = \mathfrak{h} \times \mathfrak{a}$$. Is there any formula for $$H^k(\mathfrak{g},\mathfrak{g})$$ in terms of $$H^p(\mathfrak{h},\mathfrak{h})$$ and $$H^q(\mathfrak{a},\mathfrak{a})$$? I am interesting in the case when $$\mathfrak{a}=\mathbb{C}$$ and $$\mathfrak{h}$$ is a Lie algebra such that $$H^p(\mathfrak{h},\mathfrak{h})=0$$ for all $$p\geq 0$$.

I only know Künneth formulas for the trivial module $$\Bbb C$$. But one could also use the Hochschild-Serre formula for the semidirect product $$\mathfrak{g}=\mathfrak{h}\ltimes \mathfrak{a}$$, which gives $$H^k(\mathfrak{g},\mathfrak{g})\cong \bigoplus_{i+j=k}H^i(\mathfrak{h},\Bbb C)\otimes H^j(\mathfrak{a},\mathfrak{g})^{\mathfrak{h}}.$$