 Algebraic torsion of a Dirac field

Use $K_{abc}=e_a{}^{\mu }(\omega -\omega (e){)}_{\mu bc}$ and $T_{abc}=K_{bac}-K_{cab}$. For the symmetrized Dirac action $-e\bar{\chi }{\gamma }^{\mu }{\overleftrightarrow{\nabla }}_{\mu }\chi /2-em\bar{\chi }\chi$, varying the spin connection gives $-e\bar{\chi }{\gamma }^{\mu ab}\chi \delta {\omega }_{\mu ab}/4$. Only totally antisymmetric contorsion is sourced. Its action is $-e{K}^2/(2{\kappa }^2)-e{K}_{abc}\bar{\chi }{\gamma }^{abc}\chi /4$, hence the stated solution. Substitution generates $e{\kappa }^2(\bar{\chi }{\gamma }^{abc}\chi )(\bar{\chi }{\gamma }_{abc}\chi )/32$. The torsion is an auxiliary field, not an independent propagating degree of freedom.