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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 $checkalpha : E rightarrow mathbb R$ by: $checkalpha(lambda) = frac{2(alpha,lambda)_E}{(alpha,alpha)_E}$.

Then $(Phi, E)$ is a root system, i.e. it satisfies:

1. $0 notin Phi, ; E = text{Span}_{mathbb R}(Phi)$




2. If $alpha, beta in Phi$ then $checkbeta(alpha) in mathbb Z$



3. 
   

   Define $omega_alpha : E rightarrow E $ by $omega_alpha(lambda) = lambda - checkalpha(lambda)alpha$.

   
   
   

   Then $alpha in Phi Rightarrow omega_alpha(Phi) = Phi$

   
   



4. If $alpha in Phi$ and $calpha in Phi$ then $c = pm 1$

Consider now $checkPhi = {checkalpha mid alpha in Phi}$ and $E^*$ the dual space of $E$. I am asked to prove that $(checkPhi, E^*)$, the dual root system, is indeed a root system.

I have shown that properties 1 and 4 hold for $(checkPhi, E^*)$ 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:

$check{checkalpha} : E^* rightarrow mathbb R$ and $omega_{checkalpha} : E^* rightarrow E^*$ for a given $checkalpha$ such that for $checklambda in E^*$ we have:

$check{check{alpha}}(checklambda) = frac{(checkalpha, checklambda)_{E^*}}{(checkalpha, checkalpha)_{E^*}}$, and $omega_{checkalpha}(checklambda) = checklambda - check{checkalpha}(checklambda)checkalpha$

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?