The group of all transformations that preserve some bilinear form, notable examples:
- orthogonal group preserves the inner product
- unitary group preserves a Hermitian form
- Lorentz group preserves the Minkowski inner product
We can almost reach the Lie algebra of any isometry group in a single go. For every in the Lie algebra we must have:
because has to be in the isometry group by definition as shown at Section "Lie algebra of a matrix Lie group".
Then:
so we reach:
With this relation, we can easily determine the Lie algebra of common isometries:
Bibliography:
