biquaternions lie algebra and complex-projectivespaces

In this paper, lie group and lie algebra structures of unit complex 3sphere mathbb{s}^3_{ mathbb{c} are studied. in order to do this, adjoint representation of unit biquaternions (complexified quaternions) is obtained. also, a correspondence between the elements of mathbb{s}^3_{ mathbb{c} and the special bicomplex unitary matrices su_{ mathbb{}^2}(2) is given by expressing biquaternions as 2dimensional bicomplex numbers mathbb{c}^2_2 . the relation so( mathbb{r}^3)= mathbb{s}^3/{ { pm 1 }}= mathbb{r}p^3 among the special orthogonal group so( mathbb{r}^3) , the quotient group of unit real quaternions mathbb{s}^3/{ { pm 1 }} and the projective space = mathbb{r}p^3 is known as the euclideanprojective space [1]. this relation is generalized to the complexprojective space and is obtained as so( mathbb{c}^3) cong mathbb{s}^3_{ mathbb{c}}/{ { pm 1 }}= mathbb{c}p^3 .