I'm having a bit of trouble with the definition of coroots.
From textbooks, we know that given a root system $\Phi$ with $\alpha \in \Phi$, there exists a coroot system $\Phi^{\vee}$ with $\alpha^{\vee} \in \Phi^{\vee}$ such that $\alpha^{\vee} = \frac{2 \alpha}{\alpha^2}$. But $\Phi$ consists of elements of $\mathfrak{g_0^*}$ whilst $\Phi$ consists of elements of $\mathfrak{g_0}$. Therefore I don't understand how $\alpha^{\vee}$ can be defined as such, given this would make it an element of $\mathfrak{g_0^*}$.
Given the standard isomorphism $\mathfrak{i}:\mathfrak{g_0} \to \mathfrak{g_0^*}$, shouldn't we take $\alpha^{\vee} = \frac{2 \mathfrak{i^{-1}}(\alpha)}{\alpha^2}$ instead?