Is there an example of a chiral Lie group?In particular, is it true to say that the map $g\mapsto g^{-1}$ is orientation reversing for odd dimensional Lie groups?
Moreover is there a concept of chiral Lie algebra, a finite dimesnional Lie algebra $L$ such that every automorphism of $L$ necessarily preserves the orientation?
The rational cohomology $H^{\bullet}(G, \mathbb{Q})$ of a compact connected Lie group is the exterior algebra on some odd generators, the product of which lives in top cohomology. The number of generators $r$ is the rank. The map $g \mapsto g^{-1}$ acts by $-1$ on each generator, and so it acts on top cohomology by $(-1)^r$. Hence $g \mapsto g^{-1}$ reverses orientations iff $r$ is odd iff $\dim G$ is odd.