Let $X_{i}, i=1,2,3,4$ being the generators of my Lie algebra. $X_{i}$ (i.e. taking the adjoint representations) generates the algebra $X=aX_{1}+bX_{2}+ cX_{3}+dX_{4}$ with four parameters. Using the Cartan-Killing form, $\kappa(X_{i},X_{i})= Trace(X_{i}^{2})$, we can check if the subalgebra having a compact or non-compact generators. Compact generator if $\kappa(X_{i},X_{i})$ is negative-definite and non-compact if $\kappa(X_{i},X_{i})$ being positive definite.
My question is, what happen if one of the subalgebra involved indefinite answers. Meaning for example, $\kappa(X_{1},X_{1})= -a^{2} + \alpha a^{2}$ for some constant alpha. Is this acceptable? If yes how do I categorized the generators?
