It is shown that every Lie algebra can be represented as a bivector algebra; hence every Lie group can be represented as a spin group. Thus, the computational power of geometric algebra is available to simplify the analysis and applications of Lie groups and Lie algebras. The spin version of the general linear group is thoroughly analyzed, and an invariant method for constructing real spin representations of other classical groups is developed. Moreover, it is demonstrated that every linear transformation can be represented as a monomial of vectors in geometric algebra.
C. J. L. Doran, D. Hestenes, F. Sommen and N. van Acker, Lie Groups as Spin Groups, J. Math. Phys. 34(8), 3642-3669 (1993)