If $G$ is a connected Lie group and $K$ is a closed subgroup of $G$ then $G/K$ is a homogeneous space. If $\frak g,k$ are the lie subalgebras of $G,K$ resp. Then under the projection $\pi:G\rightarrow G/K$ we get $\mathfrak g/\mathfrak k\cong T_o(G/K)$.
A homogeneous space is called reductive if there exists a subspace $\frak m$ of $\frak g$ such that $\frak g= k\oplus m$ and $Ad(k)\frak m\subset m$ for all $k\in K$. That is, $\frak m$ is $Ad(K)$-invariant.
My question is: How would the condition $Ad(k)\frak m\subset m$ imply that $\frak [k,m]\subset m$? And why the converse is true in case $K$ is connected?
My Attempt: Let $X\in \frak k$ and $Y\in \frak m$. Assume $[X,Y]\in \frak k$, then $exp\ t[X,Y]\subset K$. Hence, $exp\ t(ad_XY)=exp\ t(\frac d {ds}\{Ad(exp\ sX)Y\}|_{s=0})$. Since $Ad(exp\ sX)Y\in \frak m$ for all $s\in \mathbb R$ then, $\frac d {ds}\{Ad(exp\ sX)Y\}|_{s=0}\in \frak m$ since it is a subspace. Therefore, $exp\ t(ad_XY)=exp\ t Y'\subset K$ for some $Y'\in \frak m$. But this is a contradiction since there is a one-to-one correspondence between the elements in $\frak k$ and the one parameter subgroups in $K$.
Is my proof okay? Any comments would be appreciated!
The argument is not really correct. To show that $[X,Y]\in\mathfrak m$ is is not enough to show that $[X,Y]\notin\mathfrak k$. There is no need for a proof by contradiction here: Just take $X\in\mathfrak k$ and $Y\in\mathfrak m$. Then you know that $Ad(exp(tX))(Y)\in\mathfrak m$ by assumption. Compatibility of homomorphisms with exp shows that $Ad(exp(tX))(Y)=e^{tad(X)}(Y)=Y+t[X,Y]+t^2[X,[X,Y]]+\dots$ and differentiating at $t=0$ you get $[X,Y]\in\mathfrak m$.
Conversely, knowing that $[X,Y]\in\mathfrak m$, you readily conclude that $ad(X)^k(Y)\in\mathfrak m$ and hence $e^{tad(X)}(Y)=Ad(exp(tX))(Y)\in\mathfrak m$ for all $t$. Hence $Ad(k)(Y)\in\mathfrak m$ for any $k$ in the subgroup of $K$ generated by $exp(\mathfrak k)$. If $K$ is connected, then this subgroup coincides with $K$.
The result is just a special case of the result on invariance vs. infinitesimal invariance of subspaces in a representation of a Lie group.