  We study the interplay between geometrically-Bott-Chern-formal metrics and
SKT metrics. We prove that a $6$-dimensional nilmanifold endowed with a
invariant complex structure admits an SKT metric if and only if it is
geometrically-Bott-Chern-formal. We also provide some partial results in higher
dimensions for nilmanifolds endowed with a class of suitable complex
structures. Furthermore, we prove that any K\"ahler solvmanifold is
geometrically formal. Finally, we explicitly construct lattices for a complex
solvable Lie group in the list of Nakamura [23] on which we provide a non
vanishing quadruple $ABC$-Massey product.
