If $G$ is a Lie group, we define a maximal [Lie] torus in $G$ to be a maximal connected compact abelian Lie subgroup of $G$. These guys correspond to Cartan subalgebras of $\mathfrak{g}=Lie(G)$.
If $G$ is a linear algebraic group, we define a maximal [algebraic] torus to be a maximal abelian semisimple subgroup isomorphic to a power of the multiplicative group of the base field.
Suppose now I take $G$ to be linear algebraic over $\mathbb{C}$, so it is also a complex Lie group. What is the relationship between maximal Lie tori and maximal algebraic tori? How do we see algebraic tori at the Lie algebra level?
The wiki page about algebraic tori says they were introduced in "analogy with the theory of tori in Lie group theory", but I have difficulties in understanding the explicit connection, beyond the "picture" of $S^1$ inside $\mathbb{C}^*$.