Given $(e, h, f)$ the standard basis for $sl(2)$, I have to calculate the dual basis $(x, y, z)$ with respect to the Killing form. Now, following the definition, I am determining $x$ by imposing these conditions: $k(e,x) = 1$ $k(h, x)= 0$ $k(f,x) = 0$
By developing these equations, I have obtained some conditions on some coefficients of $ad(x)$, but not on all of them. What I am missing? Is this procedure correct? And if so, how do I obtain $x$ after computing all the entries of $ad(x)$?