BPS States in M Theory and Twistorial Constituents

We provide a complete algebraic description of Bogomol'nyi-Prasad-Sommerfield (BPS) states in $M$ theory in terms of primary constituents that we call BPS preons. We argue that any BPS state preserving $k$ of the 32 supersymmetries is a composite of $(32-k)$ BPS preons. In particular, the BPS states corresponding to the basic $M2\mathrm{}$ and $M5\mathrm{}$ branes are composed of 16 BPS preons. By extending the $M$ algebra to a generalized $D=11\mathrm{}$ conformal superalgebra $\mathrm{osp}\left(1\mid 64\right)$ we relate the BPS preons with its fundamental representation, the $D=11\mathrm{}$ supertwistors.