Let $G$ be a connected, reductive group over a field $k$, and $A_0$ a maximal split torus of $G$. Let $\Phi = \Phi(G,A_0)$ be the set of roots of $A_0$ in $G$. Then the $\mathbb R$-linear span $\mathfrak a_0^{G \ast}$ of $\Phi$ in $\mathfrak a_0^{\ast} = X(A_0) \otimes \mathbb R$ is a root system. Then in the dual space $\mathfrak a_0^G$ of $\mathfrak a_0^{G \ast}$ we have a set $\Phi^{\vee}$ of coroots. By definition, the coroot $\alpha^{\vee}$ associated to a given root $\alpha$ is the unique linear functional on $\mathfrak a_0^{G \ast}$ such that
$$s_{\alpha}(x) := x - \langle x, \alpha^{\vee} \rangle \alpha$$ is a reflection on $\mathfrak a_0^{G \ast}$ which sends $\alpha$ to $-\alpha$ and permutes $\Phi$.
If $A_G$ is the split component of $G$, there is a canonical injection of $\mathfrak a_G^{\ast} = X(A_G) \otimes \mathbb R$ into $\mathfrak a_0^{\ast}$, such that $\mathfrak a_0^{\ast} = \mathfrak a_G^{\ast} \oplus \mathfrak a_0^{G \ast}$. We then extend each element of $\mathfrak a_0^G$ to a linear functional on all of $\mathfrak a_0^{\ast}$, by setting it to be zero on $\mathfrak a_G^{\ast}$. Thus the coroots are elements of $$\mathfrak a_0 = \operatorname{Hom}_{\mathbb R}(X(A_0) \otimes \mathbb R, \mathbb R) = \operatorname{Hom}(X(A_0),\mathbb R) = X^{\ast}(A_0)$$ When $G$ is split, there is a very nice way to describe coroots, without reference to reflections or root systems. Namely, if $\alpha \in \Phi$, and $T_{\alpha} = (\operatorname{Ker} \alpha)^0$, then $\alpha^{\vee}$ is the unique cocharacter of $A_0$ satisfying:
(i): $\langle \alpha, \alpha^{\vee} \rangle = 2$
(ii): The group generated by $T_{\alpha}$ and the image of $\alpha^{\vee}$ is all of $A_0$.
Suppose that $G$ is not split. Is there any nice way to characterize the coroot $\alpha^{\vee}$ as in the split case?