  We prove a number of results concerning the embedding of a Banach lattice $X$
into an r.i. space $Y$. For example we show that if $Y$ is an r.i. space on
$[0,\infty)$ which is $p$-convex for some $p>2$ and has nontrivial concavity
then any Banach lattice $X$ which is $r$-convex for some $r>2$ and embeds into
$Y$ must embed as a sublattice. Similar conclusions can be drawn under a
variety of hypotheses on $Y$; if $X$ is an r.i. space on $[0,1]$ one can
replace the hypotheses of $r$-convexity for some $r>2$ by $X\neq L_2.$
  We also show that if $Y$ is an order-continuous Banach lattice which contains
no complemented sublattice lattice-isomorphic to $\ell_2,$ $X$ is an
order-continuous Banach lattice so that $\ell_2$ is not complementably lattice
finitely representable in $X$ and $X$ is isomorphic to a complemented subpace
of $Y$ then $X$ is isomorphic to a complemented sublattice of $Y^N$ for some
integer $N.$
