  We give a procedure for constructing an $8n$-dimensional HKT Lie algebra
starting from a $4n$-dimensional one by using a quaternionic representation of
the latter. The strong (respectively, weak, hyper-K\"ahler, balanced) condition
is preserved by our construction. As an application of our results we obtain a
new compact HKT manifold with holonomy in $SL(n,\Bbb H)$ which is not a
nilmanifold. We find in addition new compact strong HKT manifolds.
  We also show that every K\"ahler Lie algebra equipped with a flat,
torsion-free complex connection gives rise to an HKT Lie algebra. We apply this
method to two distinguished 4-dimensional K\"ahler Lie algebras, thereby
obtaining two conformally balanced HKT metrics in dimension 8.
  Both techniques prove to be an effective tool for giving the explicit
expression of the corresponding HKT metrics.
