Suppose $K$ is a Killing vector satisfying Killing's equation, $\nabla_bK_a+\nabla_aK_b=0$. As part of a larger problem, I am wondering if I have correctly shown that $$\nabla_a\nabla_bK_c + \nabla_b\nabla_cK_a + \nabla_c\nabla_aK_b = 0$$
My present proof, however, seems too easy. First, define $$\Pi_{abc} = \nabla_a\nabla_bK_c + \nabla_b\nabla_cK_a + \nabla_c\nabla_aK_b.$$ Now if we use geodesic coordinates at a certain point (wherein $\nabla_a = \partial_a$, since $\Gamma^a_{bc}=0$), the covariant derivatives commute at this point to give $$\Pi_{abc} = \nabla_b\nabla_aK_c + \nabla_c\nabla_bK_a + \nabla_a\nabla_cK_b = \Pi_{bac}.$$ Meanwhile, since $K$ is a Killing vector, $$\Pi_{abc} = -\nabla_a\nabla_cK_b - \nabla_b\nabla_aK_c - \nabla_c\nabla_bK_a = -\Pi_{bac}$$
Combining these results, we have $\Pi_{bac} = -\Pi_{abc} = -\Pi_{bac}$, meaning $\Pi_{abc}=0$, at least at the point where we have established geodesic coordinates. But since we can establish such coordinates anywhere, $\Pi_{abc}=0$ identically. Is this sound reasoning?
