  In this paper we apply the hyper-K\"ahler quotient construction to Lie groups
with a left invariant hyper-K\"ahler structure under the action of a closed
abelian subgroup by left multiplication. This is motivated by the fact that
some known hyper-K\"ahler metrics can be recovered in this way by considering
different Lie group structures on $\H^p \times \H^q$ ($\H$: the quaternions).
We obtain new complete hyper-K\"ahler metrics on Euclidean spaces and give
their local expressions.
