Let $\mathfrak{g}$ be a semisimple Lie Algebra, $\mathfrak{t}$ a Cartan Subalgebra, $\Phi$ the roots with respect to $\mathfrak{t}$, and $E = \text{Span}_{\mathbb R}(\Phi)$ a Euclidean space with the Euclidean inner product $(\cdot, \cdot)_E$
For $\alpha \in E$ define $\check\alpha : E \rightarrow \mathbb R$ by: $\check\alpha(\lambda) = \frac{2(\alpha,\lambda)_E}{(\alpha,\alpha)_E}$.
Then $(\Phi, E)$ is a root system, i.e. it satisfies:
$0 \not\in \Phi, \; E = \text{Span}_{\mathbb R}(\Phi)$
If $\alpha, \beta \in \Phi$ then $\check\beta(\alpha) \in \mathbb Z$
Define $\omega_\alpha : E \rightarrow E $ by $\omega_\alpha(\lambda) = \lambda - \check\alpha(\lambda)\alpha$.
Then $\alpha \in \Phi \Rightarrow \omega_\alpha(\Phi) = \Phi$
If $\alpha \in \Phi$ and $c\alpha \in \Phi$ then $c = \pm 1$
Consider now $\check\Phi = \{\check\alpha \mid \alpha \in \Phi\}$ and $E^*$ the dual space of $E$. I am asked to prove that $(\check\Phi, E^*)$, the dual root system, is indeed a root system.
I have shown that properties 1 and 4 hold for $(\check\Phi, E^*)$ but I am struggling with 2 and 3.
Specifically, I am unsure of what $(\cdot, \cdot)_{E^*}$ is. I suspect that I will want to define the functions:
$\check{\check\alpha} : E^* \rightarrow \mathbb R$ and $\omega_{\check\alpha} : E^* \rightarrow E^*$ for a given $\check\alpha$ such that for $\check\lambda \in E^*$ we have:
$\check{\check{\alpha}}(\check\lambda) = \frac{(\check\alpha, \check\lambda)_{E^*}}{(\check\alpha, \check\alpha)_{E^*}}$, and $\omega_{\check\alpha}(\check\lambda) = \check\lambda - \check{\check\alpha}(\check\lambda)\check\alpha$
But I don't really understand what this means. Specifically, I suspect there ought to be some sort of relationship between $(\cdot, \cdot)_E$ and $(\cdot, \cdot)_{E^*}$ but I'm not sure what it is.
Should I instead endow $E^*$ with a specific inner product that I construct so that everything works? If I do that, how should I go about constructing one?