I would like to have a glimpse of understanding of grand unificaction and all that. But before that, of course I must understand the Spin groups! I like to explicit construct them fom Clifford algebras. Below I attached three pages where it is described what to do. Sadly, I do not understand fully what goes on, especially the subscripts and superscripts do not make to much sense. Can some one guide me through equations (2) till (6)? What does (7) mean? Are the [itex]\sigma[/itex]'s the generators? What are then the [itex]\gamma[/itex]'s? Even more and above all, I would love to explicitly construct SO(4), SO(6) and so one. Can some help me to do that? thank you EDIT: But my ultimate goal is to understand what is a spinoral and what is vector representation and how are they related. On page three of the document, the author talks about vectors and the two spinors. Why are the [itex]\sigma[/itex]'s shown to be the same for the vector and both spinor representations? Are not the [itex]\sigma[/itex]'s the representations, and the vectors and spinors what the matrix representations act on?