I'm recently confused with spinor contracting with gamma matrices. Here is an example in String Theory on SO(8): $$\psi^{AB} \underbrace{|A\rangle\otimes|B\rangle}_{8_s\otimes 8_s}=[\underbrace{A(x)(C^{-1})^{AB}}_1+\underbrace{A_{ij}(x)(\gamma^{ij} C^{-1})^{AB}}_{28}+\underbrace{A_{ijkl}(x)(\gamma^{ijkl}C^{-1})^{AB}}_{35}] |A\rangle\otimes|B\rangle$$ Numbers below are dimensions.
Another example is in SO(6) two spinor reps will make two tensor reps ($4\times4=10+6$): $$\psi^\alpha \chi^\beta=A(\Gamma_M C^{-1})^{\alpha\beta}(\psi C\Gamma_M \chi)+B(\Gamma^{MNR}C^{-1})^{\alpha\beta}\psi C\Gamma_{MNR} \chi$$ My question is, which theorem supports such decomposition, espetially from the point of view of linear algebra? Why these gamma matrices are able to form a sort of basis for the production of spinors? (Here I took the example of two spinor multiplying each other, is such decomposition possible for the product of any number of spinors?)