biquaternions lie algebra and complex-projective spaces

in this paper, lie group structure and lie algebra structure of unit complex 3-sphere     are studied. in order to do this, adjoint representations of unit biquaternions (complexified quaternions) are obtained. also, a correspondence between the elements of     and the special complex unitary matrices    (2) is given by expressing biquaternions as 2-dimensional bicomplex numbers    .  the relation among the special orthogonal group       , the quotient group of unit real quaternions         and the projective space     given as                         is known as the euclidean-projective space [toth g. glimpses of algebra and geometry. springer-verlag; 1998]. this relation is generalized to the complex-projective space and is obtained as                           .