  A quantum spin-1/2, and its associated su(2) algebra of Pauli spin matrices
are familiarly linked to Clifford algebra and quaternions. Somewhat more
loosely, we develop connections between the su(4) algebra of two spins and of
its sub-algebras, which are important throughout the field of quantum
information, with octonions, the projective plane of seven elements, and
entities in projective geometry and design theory called partial Steiner
systems of 10 and 15 elements. Although not the exact correspondence between a
single spin and quaternions, the close analogy between these geometrical
objects and the 15 operators of a two-spin system may be useful to both
projective geometry and quantum information.
