Andrius: I'm writing here some notes on background information for understanding spin representations.
Background knowledge for Understanding Spin Representations
Interior product
I found helpful
I have trouble understanding
Exterior derivative
The second example in the Wikipedia article is very helpful.
Geometric product
The geometric product is just the Clifford product applied to (pure) vectors. Suppose {$e_1^2=e_2^2=1$}.
{$(x_1e_1 + x_2e_2)(y_1e1 + y_2e_2) = (x_1y_1 + x_2y_2) + (x_1y_2 - x_2y_1)e_1e_2$}
So we see that it breaks down into the dot product and the wedge product. In general, multiplying monomials increase the degree to the extent that they have different generators, and decrease the degree to the extent that they have the same generators.
Math Stack Exchange. Visualizing the area described by the dot product?
{$(\textbf{v}\cdot\textbf{w})^2 + \parallel \textbf{v}\wedge\textbf{w} \parallel^2 = \parallel \textbf{v} \parallel^2 \parallel \textbf{w} \parallel^2 $}
Spinors and zero
An isotropic vector has length zero but can have an orientation.
Odd orthogonal group has an outer zero and even orthogonal group has an inner zero (this latter case yields chiral spinors)