  We solve a problem mentioned in an article of Berger and Bourn: we prove that
in the context of an algebraically coherent semi-abelian category, two natural
definitions of the lower central series coincide. In a first, "standard"
approach, nilpotency is defined as in group theory via nested binary
commutators of the form $[[X,X],X]$. In a second approach, higher Higgins
commutators of the form $[X,X,X]$ are used to define nilpotent objects. The two
are known to be different in general; for instance, in the context of loops,
the definition of Bruck is of the former kind, while the commutator-associator
filtration of Mostovoy and his co-authors is of the latter type. Another
example, in the context of Moufang loops, is given in Berger and Bourn's paper.
In this article, we show that the two streams of development agree in any
algebraically coherent semi-abelian category. Such are, for instance, all
Orzech categories of interest. Our proof of this result is based on a
higher-order version of the Three Subobjects Lemma of Cigoli-Gray-Van der
Linden, which extends the classical Three Subgroups Lemma from group theory to
categorical algebra. It says that any $n$-fold Higgins commutator $[K_1,
\dots,K_n]$ of normal subobjects $K_i$ of an object $X$ may be decomposed into
a join of nested binary commutators.
