Split Casimir Operator of the Lie Algebra so(2r) in Spinor Representations, Colour Factors and Yang-Baxter Equation

In this paper, we derive characteristic identities for the split Casimir operator of the Lie algebra $so(2r)$ in tensor products of spinor representations of the same and opposite chiralities. Using these identities, we explicitly construct projectors onto invariant subspaces of this operator and compute their traces. The results obtained allow us to derive explicit expressions for the colour factors of ladder Feynman diagrams in gauge theories with gauge group $Spin(2r)$. In addition, we obtain a new form of a solution to the Yang-Baxter equation that is invariant under the action of the Lie algebra $so(2r)$ in spinor representations.