Multiple polylogarithms in weight 4

We clarify the relationship between different multiple polylogarithms in weight~4 by writing suitable linear combinations of a given type of iterated integral I_{n_1,…,n_d}(z_1,…,z_d), in depth d>1 and weight \sum_i n_i=4 in terms of the classical tetralogarithm Li_4. In the process, we prove a statement conjectured by Goncharov which can be rephrased as writing the sum of iterated integrals I_{3,1}(V(x,y),z), where V(x,y) denotes a formal version of the five term relation for the dilogarithm, in terms of Li_4-terms (we need 122 such).