On the Grassmann T-space

We study the Grassmann T-space, S-3, generated by the commutator [x(1), x(2), x(3)] in the free unital associative algebra K < x(1), x(2) , . . .> over a field of characteristic zero. We prove that S-3 = S-2 boolean AND T-3, where S-2 is the commutator T-space generated by [x(1), x(2)] and T-3 is the Grassmann T-ideal generated by S-3. We also construct an explicit basis for each vector space S-3 boolean AND P-n, where P-n represents the space of all multilinear polynomials of degree n in x(1) , . . . , x(n), and deduce the recursive vector space decomposition T-3 boolean AND P-n = (S-3 boolean AND P-n). (T-3 boolean AND Pn-1) x(n).