I want to know how to define a Lie group that is also a Riemannian manifold and has a group operation that is anticommutative, $a * b = -b * a$ for all elements $a$ and $b$ in the group $(G, *)$.
I know that a Lie group is a smooth manifold that also has a group structure, such that the group operations of multiplication and inversion are smooth maps. I also know that we can define a bi-invariant riemannian metric on it (if it's compact). But how can I combine these two concepts and also impose the anticommutativity condition?
Is there a general way to construct such a Lie group? Or are there any examples of such a Lie group that are known or studied in the literature? Thank you.
The OP clarified in a comment that by "anticommutative" he means "noncommutative". If so, the simplest example of a noncommutative Lie group is the 3-sphere $S^3$. A natural product structure arises from viewing the 3-sphere as the unit sphere in the quaternions $\mathbb H$. Here $H=\mathbb R^4$ as a real vector space, but the point is the existence of a noncommutative product operation among elements of $\mathbb H$. Thus, let $1,i,j,k$ denote the standard basis. We define the product of quaternions to be real-linear, and also satisfy the relations $ij=-ji, jk=-kj, ki=-ik$, as well as $i^2=j^2=k^2=-1$. Note that $i,j\in S^3$ and they visibly don't commute.
I see now that there is a new edit which re-insists on anticommutativity. This is confused because the unit element in the group commutes with every other element.