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The odd-dimensional rotation groups do not contain the central inversion and are [[simple group]]s.
The even-dimensional rotation groups do contain the central inversion {{math|−''I''}} and have the group {{nowrap|1=C<sub>2</sub> = <nowiki>{</nowiki>{{math|''I''}}, {{math|−''I''}}<nowiki>}</nowiki>}} as their [[center of a group|centre]].
SO(4) is different: there is no [[Conjugation of isometries in Euclidean space|conjugation]] by any element of SO(4) that transforms left- and right-isoclinic rotations into each other. [[Reflection (mathematics)|Reflection]]s transform a left-isoclinic rotation into a right-isoclinic one by conjugation, and vice versa. This implies that under the group O(4) of ''all'' isometries with fixed point {{mvar|O}} the distinct subgroups {{math|''S''<sup>3</sup><sub>L</sub>}} and {{math|''S''<sup>3</sup><sub>R</sub>}} are
==Algebra of 4D rotations==
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