I'm studying homogeneous spaces from the book of Arvanitoyeorgos, "An introduction to Lie groups and the geometry of homogeneous spaces", but I have a doubt at page 79 (the section about the Riemannian connection).
Consider $M\cong G/H$ a homogeneous reductive ($\mathfrak{g}=\mathfrak{h}+\mathfrak{m}$) space, and $\pi_{\star,e}:\mathfrak{g}\rightarrow T_o(G/H)$ (where $\mathfrak{m}\cong T_o(G/H)$) the differential of the canonical projection.
Now, given $X\in \mathfrak{g}$, $\pi_{\star}(X)=X^{\star}_o=\pi_{\star}(X_{\mathfrak{m}})$ (the $\mathfrak{m}$ component), and the author says immediately that $X^{\star}$ is a Killing vector field... now, I don't know much about killing vector fiels apart from the definition, and since this book is quite self-contained I was wondering if there is a simple way to see why this statement should be true, because it is obscure to me. Thanks in advance.