  We begin a discussion about the maximal containments of lower central series
ideals: ideals generated by products of two-sided ideals of the lower central
series of the free associative algebra on $n$ generators. We introduce two new
ideas to the topic, the PBW grading and the pigeonhole principle, that help us
give a complete classification of the containments for $n=2$ and obtain partial
results in the general case.
