i read in mark wildon book , an introduction to lie algebras, in page 22 say that : Suppose that dim $L$ = 3 and dim $L'$ = 2. We shall see that, over $\mathbb{C}$ at least, there are infinitely many non-isomorphic such Lie Algebras. Take a basis of $L'$, say $\{y,z\}$ and extend it to a basis of $L$, say by $x$. To understand the Lie algebra $L$, we need to understand the structure of $L'$ as a Lie algebra in its own right and how the linear map $ad x : L \rightarrow L$ acts on $L'$.
I dont understand this part "To understand the Lie algebra $L$, we need to understand the structure of $L'$ as a Lie algebra in its own right and how the linear map $ad x : L \rightarrow L$ acts on $L'$."
Can someone explain more about this part, how can we understand Lie algebra $L$ with the linear map $ad x : L \rightarrow L$ acts on $L'$? and for the case take in book, only case $x \notin L'$ , why we didnt consider $x \in L'$?
Thanks for your kindness
i have problem now to prove part b of lemma 3. $ad x : L' \rightarrow L'$. i stuck to show $ad x : L' \rightarrow L$ is homomorphism. How to show $ad x([y,z]) = [ad x(y), adx(z)]$ where $y,z \in L'$.