Let $G$ be a split reductive group over a field of characteristic zero. Is there always an isogeny $G \to H \times T$ with $H$ semi-simple and $T$ a split torus ?

I have in mind the case of $\text{GL}_n$ where we have (if I am not mistaken) the isogeny : $$\text{GL}_n \to \text{PGL}_n \times \mathbf{G}_m,\;\; g \mapsto (\overline{g},\det(g)).$$

Maybe it's even true that there is always a morphism $G \to G^{ad} \times Z(G)$ but I only need the less precise result I asked above.