How to calculate ad(X)? I have looked everywhere, and maybe its really simple, and I am just being stupid, but I really don't know how to calculate ad(X). I understand that ad_x(y)=[x,y], but i just want to calculate ad(x)? I also know that Ad(g)(X) = g^(-1)Xg. "g inverse multiplied by X multiplied by g", but the determinant for my g is 0, so it can't have an inverse, hence why I can't do it this way.
My g is \begin{bmatrix}0&x&y\\x&0&z\\y&-z&0\end{bmatrix}And I have to work out ad(x1), where x1 is one of the basis of the g. I already have the basis, it is \begin{bmatrix}0&1&0\\1&0&0\\0&0&0\end{bmatrix} Thank you.

If you already know structure constants in the base you're using, then the coefficients of the adjoint action are easily computed as
$$(ad(e_i))^k_j=c_{ij}^k,$$
since
$$ad(e_i)(e_j)=[e_i, e_j]=\sum{c_{ij}^k}e_k.$$
