  We study the fine structure of the sets of involutions avoiding either 4312
(I(4321)) or 3412 (I(3412)), connecting the point of view of the decomposition
theorems with the one of the associated labelled Motzkin paths. The algebraic
generating function of the simple involutions in I(4321) is given, together
with other generating functions, while the set I(3412) is shown containing no
simple involutions of length n>2. The reverse-complement bijection maintains
the fine structures of I(4321) and trivially of I(3412).
