isotropy representation

Let $\lag$ be a Lie algebra, and $\lah\subset\lag$ a subalgebra. The isotropy representation of $\lah$ relative to $\lag$ is the naturally defined action of $\lah$ on the quotient vector space $\lag/\lah$ .

Here is a synopsis of the technical details. As is customary, we will use $$b+\lah,\, b\in\lag$$ to denote the coset elements of $\lag/\lah$ . Let $a\in\lah$ be given. Since $\lah$ is invariant with respect to $\ad_\lag(a)$ , the adjoint action factors through the quotient to give a well defined endomorphism of $\lag/\lah$ . The action is given by $$b+\lah \mapsto [a,b]+\lah,\quad b\in\lag.$$ This is the action alluded to in the first paragraph.