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We study the Dirac spectrum on compact Riemannian spin manifolds $M$ equippedwith a metric connection $\nabla$ with skew torsion $T\in\Lambda^3 M$ in thesituation where the tangent bundle splits under the holonomy of $\nabla$ andthe torsion of $\nabla$ is of `split' type. We prove an optimal lower bound forthe first eigenvalue of the Dirac operator with torsion that generalizesFriedrich's classical Riemannian estimate.

A note on generalized Dirac eigenvalues for split holonomy and torsion