Using a construction derived from the descending central series of the free groups, we produce filtrations by infinite loop spaces of the classical infinite loop spaces BSU, BU , BSO, BO, BSp, BGL 1 .R/ C and Q 0 .S 0 /.We show that these infinite loop spaces are the zero spaces of nonunital E 1 -ring spectra.We introduce the notion of q -nilpotent K-theory of a CW-complex X for any q 2, which extends the notion of commutative K-theory defined by Adem and Gómez, and show that it is represented by Z B.q; U /, where B.q; U / is the q th term of the aforementioned filtration of BU .For the proof we introduce an alternative way of associating an infinite loop space to a commutative I -monoid and give criteria for when it can be identified with the plus construction on the associated limit space.Furthermore, we introduce the notion of a commutative I -rig and show that they give rise to nonunital E 1 -ring spectra.