  We explore the double copy for self-dual gauge and gravitational fields on
self-dual background spacetimes. We consider backgrounds associated to
solutions of the second Plebanski equation and describe results with different
gauge-fixing conditions. Finally we discuss the kinematic and $w$-algebras and
the double copy, identifying modified Poisson structures and kinematic
structure constants in the presence of the self-dual background. The self-dual
plane wave and Eguchi-Hanson spacetimes are studied as examples and their
respective $w$-algebras derived.
