  Let $M$ be a $\lambda$-indexed (that is, Jensen indexed) premouse. We prove
that $M$ is iterable with respect to standard $\lambda$-iteration rules iff $M$
is iterable with respect to a natural version of Mitchell-Steel iteration
rules. Using this equivalence, we describe a background construction for
$\lambda$-indexed mice, analogous to traditional background constructions for
Mitchell-Steel indexed mice, and which absorbs Woodin cardinals from the
background universe.
  We also prove some facts regarding the correspondence between standard
iteration trees and u-iteration trees on premice with Mitchell-Steel indexing.
