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Geometric algebra is a mathematical structure that is inherent in any metricvector space, and defined by the requirement that the metric tensor is given bythe scalar part of the product of vectors. It provides a natural framework inwhich to represent the classical groups as subgroups of rotation groups, andsimilarly their Lie algebras. In this article we show how the geometric algebraof a six-dimensional real Euclidean vector space naturally allows one toconstruct the special unitary group on a two-qubit (quantum bit) Hilbert space,in a fashion similar to that used in the well-established Bloch sphere modelfor a single qubit. This is then used to illustrate the Cartan decompositionsand subalgebras of the four-dimensional special unitary group, which haverecently been used by J. Zhang, J. Vala, S. Sastry and K. B. Whaley [Phys. Rev.A 67, 042313, 2003] to study the entangling capabilities of two-qubitunitaries.Comment: 14 pages, 2 figures, in press (Proceedings of SPIE Conference on  Defense & Security

A Bloch-sphere-type model for two qubits in the geometric algebra of a 6D Euclidean vector space