Pointwise multipliers for Triebel--Lizorkin and Besov spaces on Lie groups

On a general Lie group $G$ endowed with a sub-Riemannian structure and of local dimension $d$, we characterize the pointwise multipliers of Triebel–Lizorkin spaces $F_{\alpha }^{p,q}$ for $p,q\in (1,\mathrm{\infty })$ and $\alpha >d\mathrm{/}p$, and those of Besov spaces $B_{\alpha }^{p,q}$ for $q\in [1,\mathrm{\infty }]$, $p>d$ and $d\mathrm{/}p<\alpha <1$. When $G$ is stratified, we extend the latter characterization to all $p,q\in [1,\mathrm{\infty }]$ and $\alpha >d\mathrm{/}p$.