  Let $\mathcal{L}(H)$ be the $*$-algebra of all bounded operators on an
infinite dimensional Hilbert space $H$ and let $(\mathcal{I},
\|\cdot\|_{\mathcal{I}})$ be an ideal in $\mathcal{L}(H)$ equipped with a
Banach norm which is distinct from the Schatten-von Neumann ideal
$\mathcal{L}_p(\mathcal{H})$, $1\leq p<2$. We prove that $\mathcal{I}$
isomorphically embeds into an $L_p$-space $\mathcal{L}_p(\mathcal{R}),$ $1\leq
p<2,$ (here, $\mathcal{R}$ is the hyperfinite II$_1$-factor) if its commutative
core (that is, Calkin space for $\mathcal{I}$) isomorphically embeds into
$L_p(0,1).$ Furthermore, we prove that an Orlicz ideal
$\mathcal{L}_M(H)\neq\mathcal{L}_p(H)$ isomorphically embeds into
$\mathcal{L}_p(\mathcal{R}),$ $1\leq p<2,$ if and only if it is an
interpolation space for the Banach couple
$(\mathcal{L}_p(H),\mathcal{L}_2(H)).$ Finally, we consider isomorphic
embeddings of $(\mathcal{I}, \|\cdot\|_{\mathcal{I}})$ into $L_p$-spaces
associated with arbitrary finite von Neumann algebras.
