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A sequel to \cite{gW05}, we address again the single commutator problem\cite{PT71} of Pearcy and Topping: Is every compact operator a singlecommutator of compact operators? by focusing on a 35 year old test question forthis posed in 1976 by the last named author and others: Are there any strictlypositive operators that are single commutators of compact operators? The latterwe settle here affirmatively with a modest modification of Anderson'sfundamental construction \cite{jA77} constructing compact operators whosecommutator is a rank one projection. Moreover we provide here a rich class ofsuch strictly positive operators that are commutators of compact operators andpose a question for the rest.  We explain also how these methods are related to the study of staircasematrix forms, their equivalent block tri-diagonal forms, and commutatorproblems. In particular, we present the original test question and solutionthat led to the negative solution of the Pearcy-Topping question on whether ornot every trace class trace zero operator was a commutator (or linearcombination of commutators) of Hilbert-Schmidt operators. And we show how thisevolved from staircase form considerations along with a Larry Brown result ontrace connections to ideals \cite{lB94} which itself is at the core of\cite[Section 7]{DFWW}.  The omission in \cite{gW05} of this important 35 year old test question wasinadvertent and we correct that in this paper. This sequel starts where [ibid]left off but can be read independently of [ibid].  The present paper also has a section on self-commutator equations $[X^*,X]=A$within the framework of some classical operator Lie algebras. That problem wassolved by Fan and Fong (1980) for the full algebra of compact operators, and wesolve it here for the complex symplectic Lie algebra of compact operators andfor complex semisimple Lie algebras.

$B(H)$-Commutators: A Historical Survey II and recent advances on commutators of compact operators