  This paper deals with the exponential separation of type II, an important
concept for random systems of differential equations with delay, introduced in
\JM\ et al.~\cite{MiNoOb1}. Two different approaches to its existence are
presented. The state space $X$ will be a separable ordered Banach space with
$\dim X\geq 2$, dual space $X^{*}$ and positive cone $X^+$ normal and
reproducing. In both cases, appropriate cooperativity and irreducibility
conditions are assumed to provide a family of generalized Floquet subspaces. If
in addition $X^*$ is also separable, one obtains a exponential separation of
type II. When this is not the case, but there is an Oseledets decomposition for
the continuous semiflow, the same result holds.
  Detailed examples are given for all the situations, including also a case
where the cone is not normal.
