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Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.
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Signed permutations in $ \operatorname{SO}(n) $ and normalizing an extraspecial group
An example rotation matrix between integral octonions is this matrix:
$$\frac{1}{2}\begin{bmatrix}1 & 1 &1&0&1&0&0&0\\
-1&1&1&0&-1&0&0&0\\
-1&-1&1&0&1&0&0&0\\
0&0&0&1&0&1&-1&1\\
-1&1&-1&0&1&0&0&0\\
0& …