Maximal torus of a Lie group.

Question

I would like to ask the following question: Let $G$ be a (simple if needed) connected Lie group and $T$ 'the' maximal torus of G. suppose now that G is endowed with an other Hausdorff topology $\tau$ making it a topological group. Suppose also that $T\rightarrow G$ is a continuous homomorphism of topological groups ($G$ with a the new Hausdorff topology $\tau$ and $T$ is endowed with the standard topology of lie groups) Can we conclude that the new topology on $G$ is in fact the standard topology on $G$ making it as a Lie group ? Feel free to add more conditions on $G$. Thank you in advance.