rotations are generated by simple bivectors, with the fixed plane the dual or orthogonal to the plane of the bivector. The rotation can be said to take place about that plane, in the plane of the bivector. All other bivectors generate double rotations, with the two angles of the rotation equalling the magnitudes of the two simple bivectors the non-simple bivector is composed of. Isoclinic rotations arise when these magnitudes are equal, in which case the decomposition into two simple bivectors is not unique. Bivectors in general do not commute, but one exception is orthogonal bivectors and exponents of them. So if the bivector , where B1 and B2 are orthogonal simple bivectors, is used to generate a rotation it decomposes into two simple rotations that commute as follows: It is always possible to do this as all bivectors can be expressed as sums of orthogonal bivectors. Spacetime rotations Spacetime is a mathematical model for our universe used in special relativity. It consists of three space dimensions and one time dimension combined into a single four-dimensional space. It is naturally described using geometric algebra and bivectors, with the Euclidean metric replaced by a Minkowski metric. That algebra is identical to that of Euclidean space, except the signature is changed, so (Note the order and indices above are not universal – here e4 is the time-like dimension). The geometric algebra is Cl3,1(R), and the subspace of bivectors is ⋀2R3,1. The simple bivectors are of two types. The simple bivectors e23, e31 and e12 have negative squares and span the bivectors of the three-dimensional subspace corresponding to Euclidean space, R3. These bivectors generate ordinary rotations in R3. The simple bivectors e14, e24 and e34 have positive squares and as planes span a space dimension and the time dimension. These also generate rotations through the exponential map, but instead of trigonometric functions, hyperbolic functions are needed, which generates a rotor as follows: where Ω is the bivector (e14, etc.), identified via the metric with an antisymmetric linear transformation of R3,1. These are Lorentz boosts, expressed in a particularly compact way, using the same kind of algebra as in R3 and R4. In general all spacetime rotations are generated from bivectors through the exponential map, that is, a general rotor generated by bivector A is of the form The set of all rotations in spacetime form the Lorentz group, and from them most of