Show that if $0 → W → V → V/W → 0$ is a short exact
sequence of $\mathfrak{g}$-representations then $B_V = B_{W} + B_{V/W}$ where $\mathfrak{g}$ is a lie algebra.
$B $ is a bilinear form on $\mathfrak{g}$ such that $B(x,y)=tr(\rho(x) \rho(y))$
I don't know how to relate trace to this short sequece. I think I should use the fact that evert short exact sequece of vector spaces splits. Then, $\rho(x) \rho(y)$ can be written as sum of two matrices, one in $End(W)$ and one in $End(V/W)$? Can any one give a hint?