Varieties of groups of exponent 4

It is proved that the variety of all 4-Engel groups of exponent 4 is a maximal proper subvariety of the Burnside variety B-4, and the consequences of this are discussed for the finite basis problem for varieties of groups of exponent 4. It is proved that, for r greater than or equal to 2, the 4-Engel verbal subgroup of the r-generator Burnside group B(r, 4) is irreducible as an F(2)GL(r, 2)-module. It is observed that the variety of all 4-Engel groups of exponent 4 is insoluble, but not minimal insoluble.