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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

• $\mathrm{B}_n$ : $\vert\Lambda_{\alpha}\vert=8n-6$, for $\alpha=\pm\sum_{j=i}^n\alpha_j,~1\leq i\leq n$, and $\vert\Lambda_{\alpha}\vert=8n-10$, otherwise.

• $\mathrm{C}_n$ : $\vert\Lambda_{\alpha}\vert=4n-2$, for $\alpha=\pm\sum_{j=i}^{n-1}2\alpha_j+\alpha_n,~1\leq i\leq n-1$, and $\vert\Lambda_{\alpha}\vert=8n-6$, otherwise.

• $\mathrm{F}_4$ : $\vert\Lambda_{\alpha_1}\vert=30$ and $\vert\Lambda_{\alpha_4}\vert=42$.

• $\mathrm{G}_2$ : $\vert\Lambda_{\alpha_1}\vert=14$ and $\vert\Lambda_{\alpha_2}\vert=10$.

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 $p>3$. 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 simple root $\alpha_i\in\Delta$, we set $\Lambda_{i}=\{\beta=\sum_{j=1}^l m_j(\beta){\alpha_j}\in\Phi\vert~m_i(\beta)\neq0\}$ and $n_i$ its size. Let $c_i$ 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 $\varphi_i$ as follows, \begin{align*} \left\{\begin{array}{ll} \varphi_i(h)=h,&\text{for all } h\in\mathfrak{h},\\ \varphi_i(x_\beta)=c_i^{m_i(\beta)} x_\beta,&\text{for all } \beta=\sum_{j=1}^lm_j(\beta)x_{\alpha_j}. \end{array} \right. \end{align*} Hence \begin{align*} \operatorname{tr}(\varphi_i)&=\dim\mathfrak{h}+\sum_{\beta\in\Delta}c_i^{m_i(\beta)}\\ &=\dim\mathfrak{h}+\vert\Delta\vert-n_i+\sum_{\beta\in\Lambda_i}c_i^{m_i(\beta)}\\ &=\dim\mathfrak{s}-n_i+\sum_{\beta\in\Lambda_i}c_i^{m_i(\beta)} \end{align*} Let $\Delta=\{\alpha_1,\cdots,\alpha_l\}$ be a basis of $\Phi$, with the ordering follows Dynkin diagrams in Bourbaki. From Hasse diagrams of poset of positive root systems and by induction, one obtains

• $\mathrm{A}_l~(l\geq1)$ : for all $1\leq i\leq l$, $$\operatorname{tr}(\varphi_i)=\dim\mathfrak{s}+i\left(l+1-i\right)\left(c_i+c_i^{-1}-2\right).$$

• $\mathrm{B}_l~(l\geq2)$ : for all $1\leq i\leq l$, \begin{align*} \operatorname{tr}(\varphi_i)=\dim\mathfrak{s}-i\left(4l+1-3i\right)+i\left(2(l-i)+1\right)\left(c_i+c_i^{-1}\right)+\dfrac{(i-1)i}{2}\left(c_i^2+c_i^{-2}\right). \end{align*}

• $\mathrm{C}_l~(l\geq3)$ : for all $1\leq i\leq l-1$, \begin{align*} \left\{ \begin{array}{ll} \operatorname{tr}(\varphi_i)=\dim\mathfrak{s}-i\left(4l+1-3i\right)+2i\left(l-i\right)\left(c_i+c_i^{-1}\right)+\dfrac{i(i+1)}{2}\left(c_i^2+c_i^{-2}\right),\\ \operatorname{tr}(\varphi_l)=\dim\mathfrak{s}+\dfrac{l(l+1)}{2}\left(c_l+c_l^{-1}-2\right). \end{array} \right. \end{align*}

• $\mathrm{D}_l~(l\geq3)$ : for all $1\leq i\leq l-2$, \begin{align*} \left\{ \begin{array}{ll} \operatorname{tr}(\varphi_i)=\dim\mathfrak{s}-i\left(4l-1-3i\right)+2i(l-i)\left(c_i+c_i^{-1}\right)+\dfrac{(i-1)i}{2}\left(c_i^2+c_i^{-2}\right),\\ \operatorname{tr}(\varphi_j)=\dim\mathfrak{s}+\dfrac{(l-1)l}{2}\left(c_j+c_j^{-1}-2\right),~\text{ for }j=l-1\text{ or }l. \end{array} \right. \end{align*}

• $\mathrm{E}_6$ :
  \begin{align*} \left\{ \begin{array}{ll} \operatorname{tr}(\varphi_i)=46+16\left(c_i+c_i^{-1}\right),\text{ for }i=1\text{ or }6,\\ \operatorname{tr}(\varphi_i)=28+20\left(c_i+c_i^{-1}\right)+5\left(c_i^2+c_i^{-2}\right),\text{ for }i=3\text{ or }5,\\ \operatorname{tr}(\varphi_2)=36+20\left(c_2+c_2^{-1}\right)+\left(c_2^2+c_2^{-2}\right),\\ \operatorname{tr}(\varphi_4)=20+18\left(c_4+c_4^{-1}\right)+9\left(c_4^2+c_4^{-2}\right)+2\left(c_4^3+c_4^{-3}\right). \end{array} \right. \end{align*}

• $\mathrm{E}_7$ :
  \begin{align*} \left\{ \begin{array}{ll} \operatorname{tr}(\varphi_1)=67+32\left(c_1+c_1^{-1}\right)+\left(c_1^2+c_1^{-2}\right),\\ \operatorname{tr}(\varphi_2)=49+35\left(c_2+c_2^{-1}\right)+7\left(c_2^2+c_2^{-2}\right),\\ \operatorname{tr}(\varphi_3)=39+30\left(c_3+c_3^{-1}\right)+15\left(c_3^2+c_3^{-2}\right)+2\left(c_3^3+c_3^{-3}\right)\\ \operatorname{tr}(\varphi_4)=27+24\left(c_4+c_4^{-1}\right)+18\left(c_4^2+c_4^{-2}\right)+8\left(c_4^3+c_4^{-3}\right)+3\left(c_4^4+c_4^{-4}\right),\\ \operatorname{tr}(\varphi_5)=33+30\left(c_5+c_5^{-1}\right)+15\left(c_5^2+c_5^{-2}\right)+5\left(c_5^3+c_5^{-3}\right),\\ \operatorname{tr}(\varphi_6)=49+32\left(c_6+c_6^{-1}\right)+10\left(c_6^2+c_6^{-2}\right),\\ \operatorname{tr}(\varphi_7)=79+27\left(c_7+c_7^{-1}\right). \end{array} \right. \end{align*}

• $\mathrm{E}_8$ :
  \begin{align*} \left\{ \begin{array}{ll} \operatorname{tr}(\varphi_1)=&92+64\left(c_1+c_1^{-1}\right)+14\left(c_1^2+c_1^{-2}\right),\\ \operatorname{tr}(\varphi_2)=&64+56\left(c_2+c_2^{-1}\right)+28\left(c_2^2+c_2^{-2}\right)+8\left(c_2^3+c_2^{-3}\right),\\ \operatorname{tr}(\varphi_3)=&52+42\left(c_3+c_3^{-1}\right)+35\left(c_3^2+c_3^{-2}\right)+14\left(c_3^3+c_3^{-3}\right)+7\left(c_3^4+c_3^{-4}\right),\\ \operatorname{tr}(\varphi_4)=&36+30\left(c_4+c_4^{-1}\right)+30\left(c_4^2+c_4^{-2}\right)+20\left(c_4^3+c_4^{-3}\right)\\ &\hspace{.88cm}+15\left(c_4^4+c_4^{-4}\right)+6\left(c_4^5+c_4^{-5}\right)+5\left(c_4^6+c_4^{-6}\right),\\ \operatorname{tr}(\varphi_5)=&40+40\left(c_5+c_5^{-1}\right)+30\left(c_5^2+c_5^{-2}\right)+20\left(c_5^4+c_5^{-3}\right)+10\left(c_5^4+c_5^{-4}\right)+4\left(c_5^5+c_5^{-5}\right),\\ \operatorname{tr}(\varphi_6)=&54+48\left(c_6+c_6^{-1}\right)+30\left(c_6^2+c_6^{-2}\right)+16\left(c_6^3+c_6^{-3}\right)+3\left(c_6^4+c_6^{-4}\right),\\ \operatorname{tr}(\varphi_7)=&82+54\left(c_7+c_7^{-1}\right)+27\left(c_7^2+c_7^{-2}\right)+2\left(c_7^3+c_7^{-3}\right),\\ \operatorname{tr}(\varphi_8)=&134+56\left(c_8+c_8^{-1}\right)+\left(c_8^2+c_8^{-2}\right). \end{array} \right. \end{align*}

• $\mathrm{F}_4$ :
  \begin{align*} \left\{ \begin{array}{ll} \operatorname{tr}(\varphi_1)=22+14\left(c_1+c_1^{-1}\right)+\left(c_1^2+c_1^{-2}\right),\\ \operatorname{tr}(\varphi_2)=12+12\left(c_2+c_2^{-1}\right)+6\left(c_2^2+c_2^{-2}\right)+2\left(c_2^3+c_2^{-3}\right),\\ \operatorname{tr}(\varphi_3)=12+6\left(c_3+c_3^{-1}\right)+9\left(c_3^2+c_3^{-2}\right)+2\left(c_3^3+c_3^{-3}\right)+3\left(c_3^4+c_3^{-4}\right),\\ \operatorname{tr}(\varphi_4)=22+8\left(c_4+c_4^{-1}\right)+7\left(c_4^2+c_4^{-2}\right). \end{array} \right. \end{align*}

• $\mathrm{G}_2$ :
  \begin{align*} \left\{ \begin{array}{ll} \operatorname{tr}(\varphi_1)=4+2\left(c_1+c_1^{-1}\right)+\left(c_1^2+c_1^{-2}\right)+2\left(c_1^3+c_1^{-3}\right),\\ \operatorname{tr}(\varphi_2)=4+4\left(c_2+c_2^{-1}\right)+\left(c_2^2+c_2^{-2}\right). \end{array} \right. \end{align*}

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

• $\mathrm{B}_n$ : $\vert\Lambda_{\alpha}\vert=8n-6$, for $\alpha=\sum_{j=i}^n\alpha_j,~1\leq i\leq n$, and $\vert\Lambda_{\alpha}\vert=8n-10$, otherwise.

• $\mathrm{C}_n$ : $\vert\Lambda_{\alpha}\vert=4n-2$, for $\alpha=\sum_{j=i}^{n-1}2\alpha_j+\alpha_n,~1\leq i\leq n-1$, and $\vert\Lambda_{\alpha}\vert=8n-6$, otherwise.

• $\mathrm{F}_4$ : $\vert\Lambda_{\alpha_1}\vert=30$ and $\vert\Lambda_{\alpha_4}\vert=42$.

• $\mathrm{G}_2$ : $\vert\Lambda_{\alpha_1}\vert=14$ and $\vert\Lambda_{\alpha_2}\vert=10$.

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

• $\mathrm{B}_n$ : $\vert\Lambda_{\alpha}\vert=8n-6$, for $\alpha=\pm\sum_{j=i}^n\alpha_j,~1\leq i\leq n$, and $\vert\Lambda_{\alpha}\vert=8n-10$, otherwise.

• $\mathrm{C}_n$ : $\vert\Lambda_{\alpha}\vert=4n-2$, for $\alpha=\pm\sum_{j=i}^{n-1}2\alpha_j+\alpha_n,~1\leq i\leq n-1$, and $\vert\Lambda_{\alpha}\vert=8n-6$, otherwise.

• $\mathrm{F}_4$ : $\vert\Lambda_{\alpha_1}\vert=30$ and $\vert\Lambda_{\alpha_4}\vert=42$.

• $\mathrm{G}_2$ : $\vert\Lambda_{\alpha_1}\vert=14$ and $\vert\Lambda_{\alpha_2}\vert=10$.

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 $\vert\Lambda_{\alpha_2}\vert=10$.

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

• $\mathrm{B}_n$ : $\vert\Lambda_{\alpha}\vert=8n-6$, for $\alpha=\sum_{j=i}^n\alpha_j,~1\leq i\leq n$, and $\vert\Lambda_{\alpha}\vert=8n-10$, otherwise.

• $\mathrm{C}_n$ : $\vert\Lambda_{\alpha}\vert=4n-2$, for $\alpha=\sum_{j=i}^{n-1}2\alpha_j+\alpha_n,~1\leq i\leq n-1$, and $\vert\Lambda_{\alpha}\vert=8n-6$, otherwise.

• $\mathrm{F}_4$ : $\vert\Lambda_{\alpha_1}\vert=30$ and $\vert\Lambda_{\alpha_4}\vert=42$.

• $\mathrm{G}_2$ : $\vert\Lambda_{\alpha_1}\vert=14$ and $\vert\Lambda_{\alpha_2}\vert=10$.