Isometry group of SO(n) and SE(n)

Question

We know that E(n) is the isometry group of euclidean space. But considering E(n) or SE(n) or SO(n) itself as a Lie group with a left/right invariant Riemannian metric, what is the isometry group of E(n),SE(n) and SO(n)?

Answer 1

The isometry group of $SO(n)$ (with the metric induced by the spectrum, e.g., the operator norm) is well-known. It is given by all $X\mapsto O'OX^{\pm 1}O^{-1}$, $O,O'\in SO(n)$ unless $n=4$, in which case we need to extend by $C_2$, swapping (1,4) and (2,3), and (4,1) and (3,2) entries in $\mathfrak{so}(4)$.