The Cayley algebra or algebra of octonions is used to extract the square root of the classical Hamiltonian, in place of the Dirac-Clifford algebra. The resulting wave equation is linear and covariant and possesses a positive definite conserved probability density, and a Lagrangian. The wave function is an octonion, and may be represented as a pair of quaternions. The wave function then possesses an internal degree of freedom which corresponds to multiplication from theright by the Pauli matrices. Lorentz transformation of the wave function is effected byleft multiplication, which does not mix up isospin eigenstates. The wave equation seems appropriate for a dynamical description of an isospin doublet. The electromagnetic properties of the objects described by the wave function are those of a Dirac particle with the usual magnetic moment.