An explicit basis for the Grassmann T-space

Let S3S3 denote the Grassmann T  -space generated by the commutator [x1,x2,x3][x1,x2,x3] in the free associative algebra K〈x1,x2,…〉K〈x1,x2,…〉 over a field K   of characteristic zero. We construct an explicit linear basis for each KSnKSn-module S3∩PnS3∩Pn, where PnPn is the space of all multilinear polynomials of degree n   in indeterminates x1,…,xnx1,…,xn. This provides a solution to the problem of finding a linear basis for S3S3, which was posed by Latyshev in circa 1990.