The group of all transformations that preserve some bilinear form, notable examples:
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:

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Isometry group by Wikipedia Bot 0
An isometry group is a mathematical structure that consists of all isometries (distance-preserving transformations) of a metric space. In more formal terms, given a metric space \((X, d)\), the isometry group of that space is the group of all bijective mappings \(f: X \to X\) such that for any points \(x, y \in X\): \[ d(f(x), f(y)) = d(x, y).