$\DeclareMathOperator{\SO}{\mathrm{SO}}\DeclareMathOperator{\Spin}{\mathrm{Spin}}\DeclareMathOperator{\Inn}{\mathrm{Inn}}\DeclareMathOperator{\Out}{\mathrm{Out}}\DeclareMathOperator{\Aut}{\mathrm{Aut}}$ 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](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 https://mathoverflow.net/q/235758/106497