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 $\gamma:\mathbf{R}^{\times}\rightarrow GL_2(\mathbf{R})$:
$$
t\mapsto
\left(
\begin{array}{cc}
1 & \log|t| \newline
0 & 1
\end{array}
\right)
$$