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For the Killing vector equation, I sometimes see it written in terms of spin connection $\omega$ and other times in terms of the affine connection $\Gamma$.

More clearly $$\nabla_{\mu}V_{\nu}+\nabla_{\nu}V_{\mu}=0=\partial_{\mu}V_{\nu}+\Gamma_{\mu\nu\lambda}V^{\lambda}+\partial_{\nu}V_{\mu}+\Gamma_{\nu\mu\lambda}V^{\lambda}$$

Another time, you find it satisfying this $$\partial_A V_B+\partial_B V_A=\omega_{A,CB}V^C+\omega_{B,CA}V^C$$

What is the difference between the two descriptions?

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Comments to the question (v3):

  1. The vielbein $e^A{}_{\mu}$ in the Cartan formalism is an intertwiner $$\tag{1} g_{\mu\nu}~=~e^A{}_{\mu} ~\eta_{AB} ~e^B{}_{\nu} $$ between the curved (pseudo) Riemannian metric $g_{\mu\nu}$ and the corresponding flat metric $\eta_{AB}$. Here Greek indices $\mu,\nu,\lambda, \ldots,$ are so-called curved indices, while capital Roman indices $A,B,C, \ldots,$ are so-called flat indices. See also this related Phys.SE post.

  2. The vielbein $e^A{}_{\mu}$ translates the vector field $$V~=~V^{\mu}\partial_{\mu}$$ with curved components $V^{\mu}$ into flat components $$V^A~:=~e^A{}_{\mu} V^{\mu}.$$

  3. The christoffel symbols $\Gamma^{\lambda,\mu\nu}$ and the spin connection $\omega^{AB}_{\mu}$ can be written in terms of the vielbein $e^A{}_{\mu}$, see Wikipedia for details.

  4. TL;DR: Using the vielbein $e^A{}_{\mu}$ as an intertwiner, it is possible to show that OP's various conditions for a Killing vector field are equivalent.

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