(e14, e23). There are four distinct ways of picking one bivector from each of the first two pairs, and once these first two are picked their sum yields the third bivector from the other pair. For example, (e12, e13, e14) and (e23, e24, e34). Simple bivectors in 4D In four dimensions bivectors are generated by the exterior product of vectors in R4, but with one important difference from R3 and R2. In four dimensions not all bivectors are simple. There are bivectors such as that cannot be generated by the exterior product of two vectors. This also means they do not have a real, that is scalar, square. In this case The element e1234 is the pseudoscalar in Cl4, distinct from the scalar, so the square is non-scalar. All bivectors in four dimensions can be generated using at most two exterior products and four vectors. The above bivector can be written as Similarly, every bivector can be written as the sum of two simple bivectors. It is useful to choose two orthogonal bivectors for this, and this is always possible to do. Moreover, for a generic bivector the choice of simple bivectors is unique, that is, there is only one way to decompose into orthogonal bivectors; the only exception is when the two orthogonal bivectors have equal magnitudes (as in the above example): in this case the decomposition is not unique. The decomposition is always unique in the case of simple bivectors, with the added bonus that one of the orthogonal parts is zero. Rotations in R4 As in three dimensions bivectors in four dimension generate rotations through the exponential map, and all rotations can be generated this way. As in three dimensions if B is a bivector then the rotor R is eB/2 and rotations are generated in the same way: The rotations generated are more complex though. They can be categorised as follows: simple rotations are those that fix a plane in 4D, and rotate by an angle "about" this plane. double rotations have only one fixed point, the origin, and rotate through two angles about two orthogonal planes. In general the angles are different and the planes are uniquely specified isoclinic rotations are double rotations where the angles of rotation are equal. In this case the planes about which the rotation is taking place are not unique. These are generated by bivectors in a straightforward way. Simple