Symplectic embeddings of 4-manifolds via Lefschetz fibrations

In this article we study proper symplectic and iso-symplectic embeddings of $4$–manifolds in $6$–manifolds. We show that a closed orientable smooth $4$–manifold admitting a Lefschetz fibration over $\text{C}{P}^1$ admits a symplectic embedding in the symplectic manifold $(\C P^1 \times \C P^1 \times \C P^1, \omega_{pr}),$where$\omega_{pr}$istheproductsymplecticformon$\C P^1 \times \C P^1 \times \C P^1.$ We also show that there exists a sub-critical Weinstein $6$–manifold in which all finite type Weinstein $4$–manifolds admit iso-symplectic embeddings.