I am reading a text on which the following situation appears:
Let $G$ be a a Lie group and $H$ a compact Lie subgroup. Let $\mathfrak{g}$ and $\mathfrak{h}$ be their Lie algebras respectively. The text says it is possible to find an $\mathrm{ad}\mathfrak{h}$-invariant splitting $\mathfrak{g}=\mathfrak{h}\oplus \mathfrak{p}$, that is a splitting such that $[\mathfrak{h},\mathfrak{p}]\subset \mathfrak{p}$.
As vector spaces I know it is possible to find a splitting, but how can I ensure that this splitting is invariant under the action of $\mathrm{ad}$?
I am assuming you are dealing with real or complex Lie Algebras, not on an arbitrary field.
If $H$ is compact, you can define an $\mathrm{Ad}_H$-invariant inner product on $\mathfrak g$. Since $\mathfrak h$ is a $\mathrm{Ad}_H$-invariant subspace, its orthogonal complement (with respect to this inner product) will also be $\mathrm{Ad}_H$-invariant. Taking derivatives gives your result.
Does it make any sense to you?