Let $G$ a Lie group with $\mathfrak{g}$ its Lie algebra. From now on $V,W$ will represent elements of $G$, and $u$ will be an element of $\mathfrak{g}$
Let us suppose that $\int \exp[i\,\mathrm{tr}(uV)]\,\mathrm{d}u=\delta(V)$, with $\delta(V)$ the Dirac delta distribution on $G$. Using this, I have tried to calculate $$ S := \int \exp[i\,\mathrm{tr}(uV_1)] \exp[i\,\mathrm{tr}(uV_2)] \exp[i\,\mathrm{tr}(uV_3)]\,\mathrm{d}u. $$ Perhaps, the most obvious way to proceed would be: $$ \int \exp[i\,\mathrm{tr}(uV_1)] \exp[i\,\mathrm{tr}(uV_2)] \exp[i\,\mathrm{tr}(uV_3)]\,\mathrm{d}u=\int \exp[i\,\mathrm{tr}(u(V_1+V_2+V_3))]\,\mathrm{d}u, $$ and to write $S=\delta(V_1+V_2+V_3)$, however $V_1+V_2+V_3$, even if is well defined when $V_1$, $V_2$, $V_3$ are matrices, obviously doesn't not necessarily belongs to $G$.
How can I calculate S? Perhaps it is necessary some information that I don't have. Seriously, I need some help with this. I have made a lot of juggles with this and I don't find a satisfying way to calculate it.