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For $\mathfrak{g}$ a semisimple Lie algebra, $\mathfrak{h}$ a Cartan subalgebra, $\Phi$ a root system with respect to $\mathfrak{h}$, the coroot $\check{\alpha} $ associated to a root $\alpha \in \Phi$ is usually defined as $$\check{\alpha} = \frac{2}{(\alpha,\alpha)}\alpha,$$ where $(\cdot,\cdot)$ is a bilinear form defined to be the dual of the Killing form $\kappa$, which I understand to be defined on $\mathfrak{h}^*$ by setting $(\gamma, \delta) = \kappa (t_\gamma, t_\delta)$, where $t_\gamma \in \mathfrak{h}$ for which $\gamma(X) = \kappa(t_\gamma, X)$ for all $X \in \mathfrak{h}$.

In my textbook (Goodman-Wallach) the definition of coroot is slightly different: $$ \check{\alpha} = \kappa(e_\alpha, f_\alpha) \alpha,$$ where $\{e_\alpha, f_\alpha, h_\alpha\}$ is an $\mathfrak{sl}_2$-triple in $\mathfrak{g}$. However from my working out, I have that $$ \kappa(h_\alpha, h_\alpha) = 2 \kappa (e_\alpha, f_\alpha),$$ so that on rearranging the given definition I obtain $$ \check{\alpha} = \frac{\kappa(h_\alpha,h_\alpha)}{2} \alpha = \frac{(\alpha,\alpha)}{2}\alpha.$$

So is the definition in my textbook wrong, and should we instead define $$ \check{\alpha} = \frac{1}{\kappa(e_\alpha, f_\alpha)} \alpha,$$ or have I misunderstood something in the theory?

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Half of my confusion stems from the fact that Goodman-Wallach (unlike e.g. Humphreys) do not explicitly write out what form $h_\alpha$ should take: $$ h_\alpha = \frac{2}{\kappa(t_\alpha, t_\alpha)}t_\alpha,$$ and finding appropriate $e_\alpha$ and $f_\alpha$ will still give an $\mathfrak{sl}_2$-triple. Using this explicit formula, it is easy to check that my identity $\kappa(h_\alpha, h_\alpha) = 2 \kappa (e_\alpha, f_\alpha)$ implies $$ \check{\alpha} = \frac{2}{(\alpha, \alpha)}\alpha.$$ The other half of my confusion was that in my second-last line, $\kappa(h_\alpha, h_\alpha) \neq (\alpha, \alpha)$. Once there is an explicit form for $h_\alpha$, everything else is clarified.

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