  Recently, Vatan and Williams utilize a matrix decomposition of SU(2^n)
introduced by Khaneja and Glaser to produce CNOT-efficient circuits for
arbitrary three-qubit unitary evolutions. In this note, we place the Khaneja
Glaser Decomposition (KGD) in context as a SU(2^n)=KAK decomposition by proving
that its Cartan involution is type AIII, given n is greater than 2. The
standard type AIII involution produces the Cosine-Sine Decomposition (CSD), a
well-known decomposition in numerical linear algebra which may be computed
using mature, stable algorithms. In the course of our proof that the KGD is
type AIII, we further establish the following. Khaneja and Glaser allow for a
particular degree of freedom, namely the choice of a commutative algebra a, in
their construction. Let SWAP denote a swap between qubits 1 and n. Then for
appropriate choice of a, the KGD by (SWAP v SWAP)=k1 a k2 may be recovered from
a CSD by v = (SWAP k1 SWAP) (SWAP a SWAP) (SWAP k2 SWAP).
