I have to following Lie Algebra $L=\{x\in End(\mathbb{C}^6)\colon x^tS+Sx=0\}$, where $S=[\begin{smallmatrix} 0&I_3 \\ I_3&0 \end{smallmatrix}]$, and the subalgebra $H$ given by the diagonal elements in $L$.
I have found a basis of $L$ and $H$, which are, respectively:
$B_L=\{e_{ij}-e_{i+3j+3}, 1\leq i,j\leq 3\}\cup\{e_{ij+3}-e_{ji+3}; e_{i+3j}-e_{j+3i}, 1\leq i<j\leq 3\}$,
$B_H=\{e_{ii}-e_{i+3i+3}, i=1,2,3\}$.
Now, the exercise asks me to find a basis of weight vectors for the adjoint action of $H$. I have no idea how to solve it, I guess I have to consider the weight spaces:
$L_\alpha=\{x\in L\colon [h,x]=\alpha(h)x, \text{for each $h\in H$}\}$ and then find a set of generators $x_{\alpha}$ for each $L_{\alpha}$.
The main question is, how can I find by hand such $x_\alpha$? And the functions $\alpha$? Is my idea right?
Thanks.