  We discuss $4$-dimensional achiral Lefschetz fibrations bounding
$3$-dimensional open books and study their Lefschetz fibration (LF) embedding
in a bounded $6$-dimensional manifold, in the sense of Ghanwat--Pancholi. As an
application we give another proof of the fact that every closed orientable
$4$-manifold embeds in $S^2 \times S^2 \times S^2$ . We also show that every
achiral Lefschetz fibration with hyperelliptic monodromy admits LF embedding in
$D^6 = D^2 \times D^4$ and discuss an obstruction to such LF embeddings.
