Let $G$ be be a real reductive Lie group. Is it always possible to find finitely many maximal $\mathbf{R}$-tori $\{T_i\}_{i=1}^n$ such that the group generated by the $T_i$'s is $G$.

P.S. The assumption for $G$ to be reductive is essential since $\mathbb{G}_a$ does not contain any real torus.