maximal torus by dimension count?

Question

Suppose $T$ is a maximal torus of $G$ with dimension = $n$. If there is another torus $H \subset G$ of the same dimension, could I then conclude that $H$ is also a maximal torus? In other words once you know the dimension of a maximal torus you can just use the dimension to judge whether another torus is maximal or not.

Answer 1

Yes. By definition, $H$ is contained in some maximal torus, and it's a theorem that all maximal tori are conjugate, and in particular have the same dimension. So $H$ is contained in a conjugate of $T$. Now show that if a torus is contained in another torus and they have the same dimension then they must in fact be the same.