Suppose $(W,S)$ is a Coxeter system, not necessarily irreducible. If $\alpha$ and $\beta$ are orthogonal simple roots, do their coroots $\alpha^\vee$ and $\beta^\vee$, viewed as characters $\mathbb{G}_m\to T$, for some maximal torus $T$, commute?
For ease, I was considering a Coxeter system of type $A_1\times A_1$, with simple roots $\alpha$ and $\beta$. Then the roots are $\Phi=\{\pm\alpha,\pm\beta\}$. I know that $\alpha^\vee$ takes values in $\langle X_\alpha,X_{-\alpha}\rangle$ and $\beta^\vee$ takes values in $\langle X_\beta,X_{-\beta}\rangle$, where $X_\alpha$ is the root subgroup of $\alpha$, etc.
I want to use the Chevalley commutator formula, that if $\beta\neq\pm\alpha$, then $$ [X_\alpha,X_\beta]\subseteq\prod_{i,j>0,\ i\alpha+j\beta\in\Phi}X_{i\alpha+j\beta}. $$
There are no roots of form $i\alpha+j\beta$ where $i,j>0$, so $[X_\alpha,X_\beta]=\{1\}$, and likewise for $[X_\alpha,X_{-\beta}],[X_{-\alpha},X_\beta],[X_{-\alpha},X_{-\beta}]$, so I think it is true that the coroots commute. One concern I had was if the commutator formula holds for reducible Coxeter systems.
Is this a valid argument, and is the more general case true?
The answer to the question in the title is always yes: coroots take values in a Cartan subgroup, which is abelian if the algebraic group you start with is reductive. (A related point: not every Coxeter system comes from an algebraic group, and moreover, the ones that do might come from more than one. What you really should start with is a root datum.)
But you seem to mean: If $\alpha,\beta$ are orthogonal roots, do the root subgroups $X_\alpha$ and $X_\beta$ commute?
The answer to this question is also yes, by the reasoning you suggest.