Return to Revisions

Yes. Note that every point of $M$ is a coset $gH$ for some $g\in G$, so $M$ has a special point $m_0=1\cdot H$ and $G$ has a special point $1 \in G$. You have a linear map $\mathfrak{g}=T_1 G\to T_{m_0} M$ by quotienting by $\mathfrak{h}$, the tangent to $H$, the vertical. You split this somehow, in some manner which is $H$-equivariant. That splitting is $\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{p}$. It has to be $H$-invariant because the Maurer--Cartan form transforms in the adjoint $H$-action, so splits into a sum in the two subspaces just when $\mathfrak{p}$ is $\operatorname{Ad} H$ invariant.