Edited sentence: An odd real spectral triple is a set of four objects (A, H, D, J) while an even real spectral triple is a set of five objects (A, H, D; Γ, J) such that A is a unital dense * subalgebra of a C* algebra A; H is a Hilbert space carrying a faithful representation π of A by bounded operators; D is a selfadjoint operator on H, with compact resolvent, such that for each a ∈ A, the commutator [D, π(a)] is a bounded operator on H; Γ is a selfadjoint unitary operator on H so that Γ2 = 1; J is an antilinear isometry of H onto itself