Abstract
We develop a construction of Engel structures on $4$-manifolds based on decompositions of manifolds into round handles. This allows us to show that all parallelizable $4$-manifolds admit an Engel structure. We also show that, given two Engel manifolds $M_1,M_2$ satisfying a certain condition on the characteristic foliations, there is an Engel structure on $M_1\# M_2\# (S^2\times S^2)$ which is closely related to the original Engel structures.
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