In this paper, we revisit constructions from the literature that
translate alternating automata into language-equivalent
nondeterministic automata.  Such constructions are of practical
interest in finite-state model checking, since formulas of widely used
linear-time temporal logics with future and past operators can
directly be translated into alternating automata.  We present a
construction scheme that can be instantiated for different automata
classes to translate alternating automata into language-equivalent
nondeterministic automata. The scheme emphasizes the core ingredient
of previously proposed alternation-elimination constructions, namely,
a reduction to the problem of complementing nondeterministic automata.
Furthermore, we clarify and improve previously proposed constructions
for different classes of alternating automata by recasting them as
instances of our construction scheme.  Finally, we present new
complementation constructions for 2-way nondeterministic automata from
which we then obtain novel alternation-elimination constructions.