If $\mathfrak{s}$ is a simple Lie algebra over an algebraically closed field of characteristic zero or a classical simple Lie algebra over an algebraically closed field of characteristic $p>3$, with some exception for $p$ in the simply-laced Lie algebras $\mathrm{ADE}$ : $\mathrm{A}_n~(p\vert 2n-1)$, $\mathrm{D}_n~(p\vert 4n-7)$, $\mathrm{E}_6~(p=7)$, $\mathrm{E}_7~(p=11)$ and $\mathrm{E}_8~(p=19)$. Then the trace over the automorphism group $\operatorname{Aut}(\mathfrak{s})$ is surjective.
Let $\mathfrak{h}$ be a Cartan subalgebra and $\Phi$ its root system, we have the decomposition $$\mathfrak{s}=\mathfrak{h}\oplus\bigoplus_{\alpha\in\Phi}\mathfrak{s}_\alpha$$ where $\mathfrak{s}_\alpha=\{x\in\mathfrak{s}\vert~ [h,x]=\alpha(h)x\text{ for all }h\in\mathfrak{h}\}$ is of dimension $1$. For a root $\gamma$, we set $\Lambda_\gamma^\pm=\{\beta\in\Phi\vert~\pm\gamma+\beta\in\Phi\cup\{0\}\}$ and $\Lambda_\gamma=\Lambda_\gamma^+\cup\Lambda_\gamma^-$. Let $c_\gamma$ be a nonzero scalar, with respect to the Chevalley basis $\{x_\alpha,h_i\vert~\alpha\in\Phi\text{ and }1\leq i\leq\dim\mathfrak{h}\}$, we define the automorphism $\varphi_\gamma$ as follows, \begin{align*} \left\{\begin{array}{ll} \varphi_\gamma(h)=h,&\text{for all } h\in\mathfrak{h},\\ \varphi_\gamma(x_\beta)=c_\gamma^{\pm 1} x_\beta,&\text{if } \beta\in\Lambda_\gamma^\pm, \\ %\phi(x_\alpha)=c_\alpha^{-1} x_\alpha,&~\text{if } \\ \varphi_\gamma(x_\beta)= x_\beta,&\text{otherwise}. \end{array} \right. \end{align*} Hence \begin{align*} \operatorname{tr}(\varphi_\gamma)&=\dim\mathfrak{h}+\vert\Phi\vert-\vert\Lambda_\gamma\vert+c_\gamma\vert\Lambda^+_\alpha\vert+c_\gamma^{-1}\vert\Lambda_\gamma^{-}\vert\\ &=\dim\mathfrak{s}+\left(\frac{c_\gamma+c_\gamma^{-1}}{2}-1\right)\vert\Lambda_\gamma\vert \end{align*} For all $\alpha$ in the simply-laced root systems $\mathrm{ADE}$, we have
- $\mathrm{A}_n$ : $\vert\Lambda_\alpha\vert=4n-2$.
- $\mathrm{D}_n$ : $\vert\Lambda_\alpha\vert=8n-14$.
- $\mathrm{E}_6$ : $\vert\Lambda_\alpha\vert=42$.
- $\mathrm{E}_7$ : $\vert\Lambda_\alpha\vert=66$.
- $\mathrm{E}_8$ : $\vert\Lambda_\alpha\vert=114$.
On the other hand, for $\Delta=\{\alpha_1,\cdots,\alpha_n\}$ a basis of $\Phi$, we have
$\mathrm{B}_n$ : $\vert\Lambda_{\alpha_1}\vert=8n-10$ and $\vert\Lambda_{\alpha_n}\vert=8n-6$.
$\mathrm{C}_n$ : $\vert\Lambda_{\alpha_1}\vert=8n-10$ and $\vert\Lambda_{\alpha_{n-1}}\vert=8n-6$.
$\mathrm{F}_4$ : $\vert\Lambda_{\alpha_1}\vert=30$ and $\vert\Lambda_{\alpha_4}\vert=42$.
$\mathrm{G}_2$ : $\vert\Lambda_{\alpha_1}\vert=14$ and $\vert\Lambda_{\alpha_2}\vert=10$.