Integrability of the holonomy invariant distribution

Question

Assume that $(M,g)$ is a Riemmanian manifold with holonomy group $G$. For a point $p\in M$ we decompose the tangent space $T_p M$ to irreducible $G$ invariant subspaces. This gives us natural distributions via parallel transports. Why are these distributions, integrable? This question is intended to understand the details of the answer of Prof. Bryant to the following Mo question

https://mathoverflow.net/questions/266497/parallel-transport-as-algebra-isomorphism

Answer 1

If $C\subset TM$ is one of those distributions, you have, for any $X\in\Gamma(TM)$ and any $Y\in\Gamma(C)$, $\nabla_X Y\in\Gamma(C)$, as $C$ is parallel. Now for any $X,Y\in\Gamma(C)$ you get $0=T(X,Y)=\nabla_X Y-\nabla_Y X-[X,Y]$, hence $[X,Y]\in\Gamma(C)$, i.e. $C$ is involutive.

Answer 2

Let $Z$ be a vector field perpendicular to $C$and satisfiea $\nabla ^Z_{X} =0$ then for $Y \in C$ we have $\nabla^Y_{X} \perp Z$. So it belong $C$.