A note on embedding of achiral Lefschetz fibrations

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.