  A sequel to \cite{gW05}, we address again the single commutator problem
\cite{PT71} of Pearcy and Topping: Is every compact operator a single
commutator of compact operators? by focusing on a 35 year old test question for
this posed in 1976 by the last named author and others: Are there any strictly
positive operators that are single commutators of compact operators? The latter
we settle here affirmatively with a modest modification of Anderson's
fundamental construction \cite{jA77} constructing compact operators whose
commutator is a rank one projection. Moreover we provide here a rich class of
such strictly positive operators that are commutators of compact operators and
pose a question for the rest.
  We explain also how these methods are related to the study of staircase
matrix forms, their equivalent block tri-diagonal forms, and commutator
problems. In particular, we present the original test question and solution
that led to the negative solution of the Pearcy-Topping question on whether or
not every trace class trace zero operator was a commutator (or linear
combination of commutators) of Hilbert-Schmidt operators. And we show how this
evolved from staircase form considerations along with a Larry Brown result on
trace 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 was
inadvertent 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 was
solved by Fan and Fong (1980) for the full algebra of compact operators, and we
solve it here for the complex symplectic Lie algebra of compact operators and
for complex semisimple Lie algebras.
