I've been learning about Haar measures recently. Unimodularity of a Lie group $G$ is equivalent to $|\det \text{Ad}(g)| = 1$ for all $g \in G$, where $\text{Ad}: G \to \text{GL}(\mathfrak{g})$ is the adjoint representation. When $G$ is connected this condition is equivalent to having $\text{ad}(X) = 0$ for all $X \in \mathfrak{g}$.
I've been trying to think about all the things that could go wrong if $G$ is not connected. Are there examples of Lie groups where either:
- Left translation by some element is orientation-preserving, but right multiplication is orientation-reversing, under some choice of orientation? (Such an example would justify the need for absolute value bars in the criterion above.)
- The connected component of the identity is unimodular, but $G$ is not.
The natural examples of disconnected Lie groups I know, e.g. $\text{GL}_n(\mathbb{R}), \text{O}_n(\mathbb{R})$, do not satisfy either of these. Perhaps a finite group satisfies (1) in some sort of trivial way, but I'm having a hard time thinking about what "orientation-preserving" and "orientation-reversing" mean on a $0$-dimensional space.