Maximal torus in algebraic groups Suppose that $G$ is a linear algebraic group.
We say that $T$ is a torus if $T\approx \mathbb G_m^n$.
What are the conditions on $G$ for the existence of a torus $T$ in $G$?
Thanks for any reference  for this.
 A: Every semisimple element of $G$ lies in a maximal torus of $G^{\circ}$, the connected component of $G$.  This follows from Theorem 6.4.5 (ii), Linear Algebraic Groups (2nd edition) by T.A. Springer.
In particular, if $G^{\circ}$ has a nontrivial semisimple element, then $G$ has a nontrivial torus.  On the other hand, if $G^{\circ}$ has a nontrivial torus, then it has a nontrivial semisimple element (as all the elements of a torus are semisimple).
So that's your answer: $G$ has a nontrivial torus if and only if $G^{\circ}$ has a nontrivial semisimple element.
Which connected linear algebraic groups do not have any nontrivial semisimple elements?  Those which consist entirely of unipotent elements.  Up to isomorphism, these are exactly the closed subgroups of $\operatorname{GL}_n$ whose elements are upper triangular matrices with $1$s on the diagonal (Proposition 2.4.12, Springer).
Now, what about linear algebraic groups, not necessarily connected, whose connected components consist of unipotent elements?  These aren't easy to classify as far as I know.  For example, there are groups of the form $G = G^{\circ} \times H$, where $G^{\circ}$ is connected unipotent and $H$ is finite.
