Problem
Let $L= \mathfrak{sl}(3)$ and let $M \subset L$ be the subspace of matrices of the form $\begin{pmatrix} a & b & 0 \\ c & d & 0 \\ 0 & 0 & 0 \\ \end{pmatrix}$
with $a, b, c, d \in K$ (field).
1) Prove that $M$ is a Lie subalgebra of L, isomorphic to $\mathfrak{sl}(2)$.
2) Let $L$ be a M-module, by restricting to $M$ the adjoint map $\operatorname{ad}: L \to \mathfrak{gl}(L)$. Write $L$ as a direct sum of irreducible M-modules.
3) Let $h \in \mathfrak{sl}(2)$. For every irreducible M-module of the direct sum, find the highest weight of $h$.
Attempt to a solution:
1) To prove that $M$ is isomorphic to $\mathfrak{sl}(2)$, I wrote the linear map $f: \mathfrak{sl}(2) \to M$ which associates a generic matrix in $ \mathfrak{sl}(2)$ of the form $\begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}$, to the matrix in $M$ of the form $\begin{pmatrix} a & b & 0 \\ c & d & 0 \\ 0 & 0 & 0 \\ \end{pmatrix}$
Is it correct?
2) It is known that $L$ is semisimple and therefore, by Weyl's Theorem, it is completely reducible, which explains why it is a well-posed question. Now, because $M$ is isomorphic to $ \mathfrak{sl}(2)$, we know that $L$ is a $ \mathfrak{sl}(2)$- module. Therefore, I know that $L = L_0 \bigoplus \oplus_{\alpha \in \Phi/{0}} L_{\alpha}$, where $L_\alpha= \{x \in L: ad(h)(x) = \alpha(h)x, h \in H \}$ , where $H$ is the maximal toral subalgebra of $L$, $\alpha : H \to K$ is the functional which defines the eigenvalues of the simultaneous basis of eigenvectors of the endormorphisms $ad(h)$ , with $h \in H$. You can prove from there that $H= L_0$.
How can I continue from now on?
(Notation is conforming with James Humphreys, "Introduction to Lie Algebras and Representations")