  Relationships between certain properties of maximal subalgebras of a Lie
algebra $L$ and the structure of $L$ itself have been studied by a number of
authors. Amongst the maximal subalgebras, however, some exert a greater
influence on particular results than others. Here we study properties of those
maximal subalgebras that contain Engel subalgebras, and of those that also have
codimension greater than one in $L$.
