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 https://math.stackexchange.com/q/3843014/141334 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 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?

p.s. Possible useful link but with not explicit (not enough) constructions in Automorphism group of real orthogonal Lie groups