Root spaces for a Cartan subalgebra of $\frak{sl}(2,\mathbb{C})$

Question

I'm having trouble computing this. If I take the Cartan subalgebra generated by $$\left(\begin{matrix}0&1\\-1&0\end{matrix}\right)$$ then which are the two eigenspaces for the nonzero roots?

Answer 1

In $\mathfrak{sl}(2)$ the diagonal matrices of trace zero form a Cartan subalgebra, but also the matrices $$ H=\left\{\begin{pmatrix} a & b \cr -b & a \end{pmatrix} \mid a,b \in \mathbb{C}\right\}. $$ Over the complex numbers all Cartan subalgebras are conjugated, so we may take the diagonal matrices for considering eigenspaces.