$\DeclareMathOperator{\SO}{\mathrm{SO}}\DeclareMathOperator{\Spin}{\mathrm{Spin}}\DeclareMathOperator{\Inn}{\mathrm{Inn}}\DeclareMathOperator{\Out}{\mathrm{Out}}\DeclareMathOperator{\Aut}{\mathrm{Aut}}$ How do we construct a precise map of inner + outer automorphism of special orthogonal group $\SO(n;\mathbb{R})$?
$d=2$; We can look at $\SO(2;\mathbb{R})=U(1)$ which is abelian, and we know the inner $$ \Inn(\SO(2;\mathbb{R}))=\SO(2;\mathbb{R})/Z(\SO(2;\mathbb{R}))=1 $$ $$ \Out(\SO(2;\mathbb{R}))=\mathbb{Z}/2 $$ The total $\Aut(\SO(2;\mathbb{R}))=\Inn(\SO(2;\mathbb{R})) \rtimes \Out(\SO(2;\mathbb{R}))=\mathbb{Z}/2$ We have no $\Inn(\SO(2;\mathbb{R}))$ except the identity map. I believe that we can get the $\Out(\SO(2;\mathbb{R}))=\mathbb{Z}/2$ by flipping $t \to -t$ in $$ U(1)=\{\exp(i t) | t \in [0, 2 \pi)\} \to \{\exp(-i t) | t \in [0, 2 \pi)\}. $$ I wish to see explicit answer like the above for my following questions ---
other $n$ but $n\neq 2,8$ is discussed in MSE with answer still pending.
for $n=8$
Question 1: How do we construct the inner automorphism map explicitly (if my result is correct?)? Let us consider $\Spin(8;\mathbb{R})$, $\SO(8;\mathbb{R})$, $\SO(8;\mathbb{R})/(\mathbb{Z}/2)$.
for $n=8$
$$ \Inn(\Spin(n;\mathbb{R}))=\Spin(n;\mathbb{R})/Z(\Spin(n;\mathbb{R})) = \SO(8;\mathbb{R})/\mathbb{Z}/2 $$ $$ \Inn(\SO(n;\mathbb{R})/\mathbb{Z}/2)=(\SO(n;\mathbb{R})/\mathbb{Z}/2)/Z(\SO(n;\mathbb{R})/(\mathbb{Z}/2)) = \SO(8;\mathbb{R})/\mathbb{Z}/2 $$ $$ \Inn(\SO(n;\mathbb{R}))=\SO(n;\mathbb{R})/Z(\SO(n;\mathbb{R})) = \SO(8;\mathbb{R})/\mathbb{Z}/2 $$
Question 2: How do we construct the outer automorphism map explicitly $$ \Out(Spin(8;\mathbb{R}))=S_3 $$ $$ \Out(\SO(8;\mathbb{R}))=\mathbb{Z}/2 $$ $$ \Out(\SO(8;\mathbb{R})/\mathbb{Z}/2 )=S_3 $$ Given the parametrization of $\SO(n;\mathbb{R})$ how to map to itself via the Out$\Out$ map?
Question 3: How do we construct the total automorphism map explicitly $$ \Aut(\Spin(8;\mathbb{R}))=\Inn(\Spin(8;\mathbb{R})) \rtimes \Out(\Spin(8;\mathbb{R})) =(\SO(8;\mathbb{R})/\mathbb{Z}/2 ) \rtimes S_3 ? $$ $$ \Aut(\SO(8;\mathbb{R}))=\Inn(\SO(8;\mathbb{R})) \rtimes \Out(\SO(8;\mathbb{R})) =(\SO(8;\mathbb{R})/\mathbb{Z}/2 ) \rtimes \mathbb{Z}/2 ? $$ $$ \Aut(\SO(8;\mathbb{R})/\mathbb{Z}/2)=\Inn(\SO(8;\mathbb{R})/\mathbb{Z}/2) \rtimes \Out(\SO(8;\mathbb{R})/\mathbb{Z}/2) =(\SO(8;\mathbb{R})/\mathbb{Z}/2 ) \rtimes S_3 ? $$ Given the parametrization of $\Spin(8;\mathbb{R})$, $\SO(8;\mathbb{R})$, $\SO(8;\mathbb{R})/(\mathbb{Z}/2)$, how to map to itself via the $\Aut$ map?
$$ \Aut(\Spin(8;\mathbb{R}))=\Inn(\Spin(8;\mathbb{R})) \rtimes \Out(\Spin(8;\mathbb{R})) =(\SO(8;\mathbb{R})/\mathbb{Z}/2 ) \rtimes S_3 ? $$ $$ \Aut(\SO(8;\mathbb{R}))=\Inn(\SO(8;\mathbb{R})) \rtimes \Out(\SO(8;\mathbb{R})) =(\SO(8;\mathbb{R})/\mathbb{Z}/2 ) \rtimes \mathbb{Z}/2 ? $$ $$ \Aut(\SO(8;\mathbb{R})/\mathbb{Z}/2)=\Inn(\SO(8;\mathbb{R})/\mathbb{Z}/2) \rtimes \Out(\SO(8;\mathbb{R})/\mathbb{Z}/2) =(\SO(8;\mathbb{R})/\mathbb{Z}/2 ) \rtimes S_3 ? $$ Given the parametrization of $\Spin(8;\mathbb{R})$, $\SO(8;\mathbb{R})$, $\SO(8;\mathbb{R})/(\mathbb{Z}/2)$, how to map to itself via the $\Aut$ map?
P.S. Possible useful link but with not explicit (not enough) constructions in Automorphism group of real orthogonal Lie groups
