This is a continuation of my previous question. Let $G$ be a connected reductive group over an algebraically closed field $k$ of characteristic 0. We assume that $\mathrm{Pic}\ G=0$. This is the same as to say that the derived group $G^{\mathrm{der}}$ of $G$ (which is semisimple) is simply connected. In particular, if $G$ is simply connected semisimple, then $\mathrm{Pic}\ G=0$.

Let $D$ be a diagonalizable subgroup in $G$. Is it true that $D$ is contained in some torus $T\subset G$ ?

Angelo's answer to my previous question shows that this is *not* true for $G=\mathrm{PGL}_n$.
Of course, $\mathrm{Pic}\ \mathrm{PGL}_n\neq 0$.

`$SL_n, Sp_{2n}$`

cause no trouble.) $\endgroup$ – Jim Humphreys Apr 7 '11 at 17:07