One definition of the Killing field is as those vector fields along which the Lie Derivative of the metric vanishes. But for very many calculation purposes the useful way to think of them when dealing with the Riemann-Christoffel connection for a Riemannian metric is that they satisfy the differential equation, $\nabla_\nu X_\nu + \nabla_\nu X_\nu = 0$
I suppose solving the above complicated set of coupled partial differential equations is the only way to find Killing fields given a metric. (That's the only way I have ever done it!) Would be happy to know if there are more elegant ways.
But what would be the slickest way to say prove that the scalar curvature is constant along a Killing field, i.e it will satisfy the equation $X^\mu \nabla_\mu R = 0$ ?
Sometimes it is probably good to think of Killing fields as satisfying a Helmholtz-like equation sourced by the Ricci tensor.
Then again very often one first starts off knowing what symmetries one wants on the Riemannian manifold and hence knows the Lie algebra that the Killing fields on it should satisfy. Knowing this one wants to find a Riemannian metric which has that property. As I had asked in this earlier question
There is also the issue of being able to relate Killing fields to Laplacians on homogeneous spaces and H-connection like earlier raised in this question. Good literature on this has been very hard to find beyond the very terse and cryptic appendix of a paper by Camporesi and Higuchi.
I would like to know what are the different ways of thinking about Killing fields some of which will say help easily understand the connection to Laplacians, some of which make calculations of metric from Killing fields easier and which ones give good proofs of the constancy of scalar curvature along them. It would be enlightening to see cases where the "original" definition in terms of Lie derivative is directly useful without needing to put it in coordinates.