by Proposition: If is in , then is also in . Proof: Proposition: Proof: Note first that means that the sum of the squares of its four complex components is one. Then the sum of the squares of the complex conjugates of these components is also one. Therefore, Now Associated terminology As the biquaternions have been a fixture of linear algebra since the beginnings of mathematical physics, there is an array of concepts that are illustrated or represented by biquaternion algebra. The transformation group has two parts, and The first part is characterized by ; then the Lorentz transformation corresponding to is given by since Such a transformation is a rotation by quaternion multiplication, and the collection of them is SO(3) But this subgroup of is not a normal subgroup, so no quotient group can be formed. To view it is necessary to show some subalgebra structure in the biquaternions. Let represent an element of the sphere of square roots of minus one in the real quaternion subalgebra . Then and the plane of biquaternions given by is a commutative subalgebra isomorphic to the plane of split-complex numbers. Just as the ordinary complex plane has a unit circle, has a unit hyperbola given by Just as the unit circle turns by multiplication through one of its elements, so the hyperbola turns because Hence these algebraic operators on the hyperbola are called hyperbolic versors. The unit circle in and unit hyperbola in are examples of one-parameter groups. For every square root of minus one in , there is a one-parameter group in the biquaternions given by The space of biquaternions has a natural topology through the Euclidean metric on -space. With respect to this topology, is a topological group. Moreover, it has analytic structure making it a six-parameter Lie group. Consider the subspace of bivectors . Then the exponential map takes the real vectors to and the -vectors to When equipped with the commutator, forms the Lie algebra of . Thus this study of a six-dimensional space serves to introduce the general concepts of Lie theory. When viewed in the matrix representation, is called the special linear group SL(2,C) in . Many of the concepts of special relativity are illustrated through the biquaternion structures laid out. The subspace corresponds to Minkowski space, with the four coordinates giving the time and space locations of events in a resting frame of reference. Any