Note on faithful representations and a local property of Lie groups

Let $G$ be any analytic group, let $T$ be a maximal toroid of the radical of $G$, and let $S$ be a maximal semisimple analytic subgroup of $G$. If $L=\mathcal {L}(G)$ is the Lie algebra of $G$, $\mathrm {rad}[L,L]$ is the radical of $[L,L]$, and $\mathcal {Z}(L)$ is the center of $L$, we show that $G$ has a faithful representation if and only if (i) $\mathrm {rad}[L,L]\cap \mathcal{Z}(L)\cap \mathcal{L}(T)=(0)$, and (ii) $S$ has a faithful representation.