Rotations in 4-dimensional Euclidean space

Rotations in four-dimensional Euclidean space are linear isometries that preserve orientation and fix the origin. They form the special orthogonal group SO(4).

Connect

SO(4): The group of real 4×4 orthogonal matrices with determinant 1. This matrix group is the precise algebraic form of the rotations described here.
Orthogonal group: The group of linear transformations that preserve an inner product on a real vector space. SO(4) is the determinant-one subgroup of the orthogonal group in four dimensions.
Rotations in three-dimensional Euclidean space: Orientation-preserving linear isometries of three-dimensional Euclidean space that fix the origin. A three-dimensional rotation has one fixed axis, while a generic four-dimensional rotation has two rotating planes.
Euclidean rotation: An orientation-preserving isometry of Euclidean space that fixes a point. The four-dimensional case is one instance of rotations in Euclidean spaces of different dimensions.
Isoclinic rotation: A four-dimensional rotation that turns every vector through the same angle. It is the special case in which the two invariant planes rotate equally.
Special orthogonal group: The group of orthogonal transformations with determinant 1. The rotation group in four dimensions is the dimension-four instance of this family.
Rotations in two-dimensional Euclidean space: Orientation-preserving linear isometries of the Euclidean plane that fix the origin. The plane has only one rotation angle; four dimensions allow two independent angles.
Rigid body transformation: A transformation combining rotation and translation while preserving distances. Four-dimensional rotational components can be used in rigid transformations of four-dimensional models.
Invariant subspace: A vector subspace mapped into itself by a linear transformation. Every four-dimensional rotation decomposes into invariant planes on which it acts as a planar rotation.
Eigenvalue: A scalar by which a linear transformation scales a corresponding nonzero eigenvector. Complex eigenvalues of real rotations encode their planar rotation angles.