The triality properties of Dirac spinors are studied, including a construction of the algebra of (complexified) biquaternion. It is proved that there exists a vector-representation of Dirac spinors. The massive Dirac equation in the vector-representation is actually self-dual. The Dirac’s idea of non-integrable phases is used to study the behavior of massive term.

Code loops were introduced by R. L. Griess. Griess [5] and Hsu [7] gave methods to construct the corresponding code loop from any given doubly even binary code; both these methods used some kind of induction. In this paper, we present a global construction of the loop, where we apply the correspondance between the concepts of Moufang loops and groups with triality.

We give a model for composite quarks and leptons based on the semisimple gauge group SU(4), with the preons in the 10 representation. Hypercolor singlets are forbidden in the fermionic sector of this theory; we propose that SU(4) symmetry spontaneously breaks to SU(3)× U(1), with the binding of triality nonzero preons and gluons into composites, and with the formation of a color singlet condens...

In his 2006 Cyclic Homology Course from Poland, J.L. Loday stated that the edges of the associahedron of any dimension can be labelled by elements of the Steinberg Group such that any 2-dimensional face represents a relation in the Steinberg Group. We prove his statement. We define a new group R(n) relevant in the study of the rotation distance between rooted planar binary trees .