On universal Lie nilpotent associative algebras

We study the quotient Qi(A) of a free algebra A by the ideal Mi(A) generated by the ith commutator of any elements. In particular, we completely describe such quotient for i=4 (for i⩽3 this was done previously by Feigin and Shoikhet). We also study properties of the ideals Mi(A), e.g. when Mi(A)Mj(A) is contained in Mi+j−1(A) (by a result of Gupta and Levin, it is always contained in Mi+j−2(A)).