between the vectors. It is written , where B is the bivector. For general bivectors the magnitude can be calculated by taking the norm of the bivector considered as a vector in the space ⋀2Rn. If the magnitude is zero then all the bivector's components are zero, and the bivector is the zero bivector which as an element of the geometric algebra equals the scalar zero. Unit bivectors A unit bivector is one with unit magnitude. It can be derived from any non-zero bivector by dividing the bivector by its magnitude, that is Of particular interest are the unit bivectors formed from the products of the standard basis. If ei and ej are distinct basis vectors then the product is a bivector. As the vectors are orthogonal this is just , written eij, with unit magnitude as the vectors are unit vectors. The set of all such bivectors form a basis for ⋀2Rn. For instance in four dimensions the basis for ⋀2R4 is (e1e2, e1e3, e1e4, e2e3, e2e4, e3e4) or (e12, e13, e14, e23, e24, e34). Simple bivectors The exterior product of two vectors is a bivector, but not all bivectors are exterior products of two vectors. For example, in four dimensions the bivector cannot be written as the exterior product of two vectors. A bivector that can be written as the exterior product of two vectors is simple. In two and three dimensions all bivectors are simple, but not in four or more dimensions; in four dimensions every bivector is the sum of at most two exterior products. A bivector has a real square if and only if it is simple, and only simple bivectors can be represented geometrically by an oriented plane area. Product of two bivectors The geometric product of two bivectors, A and B, is The quantity is the scalar-valued scalar product, while is the grade 4 exterior product that arises in four or more dimensions. The quantity is the bivector-valued commutator product, given by The space of bivectors ⋀2Rn is a Lie algebra over R, with the commutator product as the Lie bracket. The full geometric product of bivectors generates the even subalgebra. Of particular interest is the product of a bivector with itself. As the commutator product is antisymmetric the product simplifies to If the bivector is simple the last term is zero and the product is the scalar-valued , which can be used