I am starting to learn reductive groups, and I don't have good knowledge of the root subgroups. For a root $\alpha \in \Phi$, I do know they are define as the (unique?) one-parameter subgroup $x: G_a \to U_\alpha$ such that $$tx(a)t^{-1} = x(\alpha(t) a)$$
If I take two different root subgroups $U_\alpha$ and $U_\beta$, how can I think of the product $U_\alpha U_\beta$?
Consider matrix groups, and I did my examples in $GL(n)$. Does $U_\alpha U_\beta$ correspond to the matrices with stars (nonzero elements) at the "places" indicated by $\alpha$ and $\beta$ (e.g. for $\alpha = \epsilon_1 - \epsilon_2$ and $\beta = \epsilon_1 - \epsilon_3$) I would see that $U_\alpha$ are the matrices with $1$ on the diagonal and nonzero elements only at $(1,2)$ and $(1,3)$. In that case, they commute.
Is it always the case, or is there other recipes to think about them/construct them?