Synthematic construction of the small Witt design

On a six-point set $X$ and its six totals of synthemes $Y$, each duad determines a partition of $Y$ into three pairs according to their shared synthemes. Include both whole halves, the 45 blocks $D\cup (Y\setminus P)$ for these duad-pair incidences, and their complements. For each partition $X=A\bigsqcup A^c$ into triples, cross synthemes give two triangular components on $Y$; include the four unions of a part of $X$ with a component. These 132 blocks form the small Witt design.