  We introduce and study a fermionization procedure for the cohomological Hall
algebra $\mathcal{H}_{\Pi_Q}$ of representations of a preprojective algebra,
that selectively switches the cohomological parity of the BPS Lie algebra from
even to odd. We do so by determining the cohomological Donaldson--Thomas
invariants of central extensions of preprojective algebras studied in the work
of Etingof and Rains, via deformed dimensional reduction.
  Via the same techniques, we determine the Borel-Moore homology of the stack
of representations of the $\mu$-deformed preprojective algebra introduced by
Crawley-Boevey and Holland, for all dimension vectors. This provides a common
generalisation of the results of Crawley-Boevey and Van den Bergh on the
cohomology of smooth moduli schemes of representations of deformed
preprojective algebras, and my earlier results on the Borel-Moore homology of
the stack of representations of the undeformed preprojective algebra.
