Using for spin-³/₂ matrices a direct-product structure involving the usual Pauli spin matrices, the authors derive the Dirac-Clifford matrices in terms of certain algebraic combinations of spin-³/₂ matrices in a representation-independent way, thus achieving an extension of the Pauli spin matrices from the usual spin-¹/₂ space to the spin-³/₂ space. Basing the derivation directly on this analysis, two algebras satisfied by spin-³/₂ matrices are derived. One of these, which is also satisfied by spin-¹/₂ matrices, is directly related to the spin-³/₂ algebras of Weaver (1978) and of Bhabha and Madhava Rao (for three objects). The other algebra is new and curiously is not satisfied by spin-¹/₂ matrices.