In the wikipedia article on spinors a number of mathematical definitions are given of spinors which I find slightly confusing.
There are essentially two frameworks for viewing the notion of a spinor. One is representation theoretic. In this point of view, one knows a priori that there are some representations of the Lie algebra of the orthogonal group that cannot be formed by the usual tensor constructions. These missing representations are then labeled the spin representations, and their constituents spinors.
According to the answer by Korman of this question, every irreducible representation is found inside a tensor product of fundamental representations. So there can't be any missing representations. So how do we make sense of the above statement?
In this view, a spinor must belong to a representation of the double cover of the rotation group SO(n, R)
Whereas the statement above refers to lie algebras, this is referring to lie groups. Since there is a 1-1 correspondance between the two I guess this is fine. (But seeing that Spinors orginated in Physics around the idea of 'spin' is it better to think of Spinors as representation of lie groups rather than lie algebras?).
Representations of the double covers of these groups yield projective representations of the groups themselves, which do not meet the full definition of a representation.
Is there a 1-1 relationship between representations of double covers & projective representations of the original group?