I try to calculate the Cartan subalgebra generators in the $6$-dimensional fundamental representation of $SU(4)$ group. I'm able to find these generators however fixing the overall constants of these generators seems not possible if the trace formula $Tr(T_aT_b)=\frac{1}{2}\delta_{ab}$ is used as it does only apply to the defining representations. Using the Cartan-Killing form sounds like a probable solution as elaborated in e.g. Fuchs and Schwigert's book but I wonder if there is any other alternative that can get me to the normalization I'm looking for faster.

Any help would be appreciated.

AB