I am aware the general statement on inner / outer automorphism of special orthogonal group SO($N$). Here I am trying to show them very explicitly.
Consider SO(3) Lie algebra generators: $$ [X_i,X_j]=i \epsilon^{ijk} X_k\\ X_1=i {\begin{pmatrix} 0& 0&0\\ 0& 0&-1\\0&1&0 \end{pmatrix}}, X_2=i {\begin{pmatrix} 0& 0&1\\ 0& 0&0\\-1&0&0 \end{pmatrix}}, X_3=i {\begin{pmatrix} 0& -1&0\\ 1& 0&0\\0&0&0 \end{pmatrix}} $$
Note $X_j^\dagger=X_j$.
Write the SO(3) Lie group elements as $$ g=e^{i \theta \alpha_j X_j} $$ where $g^{-1}=g^\dagger$. And $g^{-1}g=g^\dagger g=1$. $g=e^{i \theta \alpha_j X_j}$ is the unitary Rep of SO(3) Lie group. We can choose $\sum_{j=1}^3|\alpha_j|^2=1$ so let to $\vec \alpha_j$ to be a unit vector on $S^2$.
- For SO(3) Lie group, we have an order 2 inner automorphism $\mathbb{Z}_2$, but no outer automorphisms. (Look at the symmetries of Dynkin diagram $B_n$.) How do we find the explicit $k$ such that $$ k g_{\text{SO(3)}} k^{-1} = g_{\text{SO(3)}}^{'}? $$ which is the inner automorphism arisen by the conjugation of $k$?
Similarly, I believe that we can write SO(N) Lie algebra generators: $$ [X_i,X_j]=i \epsilon^{ijk} X_k, $$ with $i,j,k \in \{1,2,..., N(N-1)/2 \}$, such that all the SO(n) Lie group elements as $$ g_{\text{SO(N)}}=e^{i \theta \alpha_j X_j}, $$ I suppose the $\theta$ has a 2$\pi$ periodicty? Then
For SO$(2n+1)$=SO$(2n+1, \mathbb{R})$ Lie group, when $2n+1$ is odd and $n \geq 1$, we have an order 2 inner automorphism $\mathbb{Z}_2$, but no outer automorphisms. (Look at the symmetries of Dynkin diagram $B_n$.) How do we find the explicit $k$ such that $$ k g_{\text{SO(2n+1)}} k^{-1} = g_{\text{SO(2n+1)}}^{'}? $$ which is the inner automorphism arisen by the conjugation of $k$?
For SO$(2n)$=SO$(2n, \mathbb{R})$ Lie group, when $2n$ is even and $n > 1$ (but not $n=1$?), we have an order 2 outer automorphism $\mathbb{Z}_2$, but no outer automorphisms. (Look at the symmetries of Dynkin diagram $D_n$.) How do we find the explicit map $$ g_{\text{SO(2n+1)}} \to g_{\text{SO(2n+1)}}^{'}? $$ which explicit map is the out automorphism?