The main result in the paper states the following: For a finite
group $G=AB$, which is the product of the soluble subgroups $A$
and $B$, if $\langle a,b \rangle$ is a metanilpotent group for all
$a\in A$ and $b\in B$, then the factor groups $\langle a,b \rangle
F(G)/F(G)$ are nilpotent, $F(G)$ denoting the Fitting subgroup of
$G$. A particular generalization of this result and some
consequences are also obtained. For instance, such a group $G$ is
proved to be soluble of nilpotent length at most $l+1$, assuming
that the factors $A$ and $B$ have nilpotent length at most $l$. Also
for any finite soluble group $G$ and $k\geq 1$, an element $g\in G$
is contained in the preimage of the hypercenter of $G/F_{k-1}(G)$,
where $F_{k-1}(G)$ denotes the ($k-1$)th term of the Fitting series
of $G$, if and only if the subgroups $\langle g,h\rangle$ have
nilpotent length at most $k$ for all $h\in G$.