Messing around in the 1-2 subgroup of the smooth mapping class group of S^4

The 1-2 subgroup of the smooth mapping class group of S^4 is the subgroup of diffeomorphisms which arise as the monodromy of the boundary of a 5-dimensional handlebody bundle over S^1 in which the handlebodies only use 0-, 1- and 2-handles and each fiber is diffeomorphic to B^5. That is a mouthful and I'll try to make sense of it for you, set in the larger context of the full smooth mapping class group of S^4, and show how to explicitly work out some relations in the 1-2 subgroup using diagrammatic tools.