$L$ acts on K via the adjoint representation

Question

I'm studing Lie Algebras using the introduction wrinten by Humphreys. I don't understand a phrases which is : $L$ acts on $K$ via the adjoint representation. (L is a lie algebra and K is an ideal of L)

To see where this accure please see the image:

Answer 1

Here $L$ acts on $K$ as follows: for $a \in L$ and $x \in K$, $a.x$ is defined to be $[a,x]$. The representation $\mbox{ad} : L \rightarrow \mbox{Der}(L) \subset \mbox{gl}(L)$ sending $x$ to $\mbox{ad}(x)$ is called the adjoint representation, where $\mbox{ad}(x)(y)=[x,y]$. Since $K$ is an ideal of $L$, we can regard $\mbox{ad} : L \rightarrow \mbox{gl}(K)$. In these reason, $L$ is said to act on $K$ via the adjoint representation. For references, see Humphreys (1.3) and (2.2).