Corollary 6

Suppose that $G$ is finite-dimensional. Let $t \in U(G)$. For $L_t$ (resp. $R_t$) to be right (resp. left) invariant, it is necessary and sufficient that $t$ belong to the centre of $U(G)$.

For $L_t$ (resp. $R_t$) to be right (resp. left) invariant, it is necessary and sufficient that $\varepsilon_g * t = t * \varepsilon_g$ for all $g \in G$, that is that $(\mathrm{Int}\, g)_* t = t$. There exists an integer $n$ such that $t \in U_n(G)$. By Corollary 3 and Proposition 45 of § 3, no. 12, $(\mathrm{Int}\, g)_* t = t$ for all $g \in G$ if and only if $[a, t] = 0$ for all $a \in L(G)$, that is if and only if $t$ commutes with $U(G)$.