Finitely based ideals of weak polynomial identities

Let K be a field, char K not equal 2, and let V-k be a k-dimensional vector space over K equipped with a nondegenerate symmetric bilinear form. Denote C-k the Clifford algebra of V-k. We study the polynomial identities for the pair (C-k, V-k). A basis of the identities for this pair is found. It is proved that they are consequences of the single identity [x(2), y] = 0 when k = infinity. It is shown that when k < infinity the identities for (C-k, V-k) follow from [x(2), y] = 0 and Wk+1 = 0 where Wk+1 is an analog of the standard polynomial St(k+1). Denote M-2(K) the matrix algebra of order two over K, and let sl(2)(K) be the Lie algebra of all traceless 2 x 2 matrices over K. As an application, new proof of the fact that the identity [x(2), y] = 0 is a basis of the weak Lie identities for the pair (M-2(K),sl(2)(K)) is given.