Let $G$ be be a connected real algebraic reductive Lie group. Is it always possible to find finitely many maximal algebraic $\mathbf{R}$-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.

added: The algebraicity assumption was added in order to avoid one-parameter subgroups like $\gamma:\mathbf{R}^{\times}\rightarrow GL_2(\mathbf{R})$: $$ t\mapsto \left( \begin{array}{cc} 1 & \log|t| \newline 0 & 1 \end{array} \right) $$