Proof of tracelessness of $\mathfrak{su}(n)$ generators

Question

When discussing special unitary groups of matrices, in all books there is a claim that is never proven. It is, that all generators of this group are traceless. On what basis is this claim made?

Answer 1

If $\xi \in \mathfrak{su}(n)$ is an infinitesimal generator (i.e. $\exp(t \xi) \in SU(n)$), then $\det \exp(t \xi) = 1$ for each $t$ and thus $\frac d{dt}|_{t = 0}\det \exp(t \xi) = 0$. Applying Jacobi's formula we can write this derivative as $$0 = \mathrm{tr}\left( \frac d{dt}\Big|_{t=0}\exp(t \xi)\right)=\mathrm{tr} (\xi).$$