  A graph is called 1-planar if it can be drawn on the plane (or on the sphere)
such that each edge is crossed at most once. A 1-planar graph $G$ is called
optimal if it satisfies $|E(G)| = 4|V(G)|-8$. If $G$ and $H$ are graphs, then
the lexicographic product $G\circ H$ has vertex set the Cartesian product
$V(G)\times V(H)$ and edge set $\{(g_1,h_1) (g_2,h_2): g_1 g_2 \in E(G),\,\,
\text{or}\,\, g_1=g_2 \,\, \text{and}\,\, h_1 h_2 \in E(H)\}$. A graph is
called reducible if it can be expressed as the lexicographic product of two
smaller non-trivial graphs. In this paper, we prove that an optimal 1-planar
graph $G$ is reducible if and only if $G$ is isomorphic to the complete
multipartite graph $K_{2,2,2,2}$. As a corollary, we prove that every reducible
1-planar graph with $n$ vertices has at most $4n-9$ edges for $n=6$ or $n\ge
9$. We also prove that this bound is tight for infinitely many values of $n$.
Additionally, we give two necessary conditions for a graph $G\circ 2K_1$ to be
1-planar.
