. Intuitively, a rotation about any axis should be an isometry. In this chart, we can immediately write down the vector field which generates rotations about the -axis: Note that in these coordinates, the metric components are all independent of , which is another way for us to see that is a Killing vector (see below). Naïvely we might hope to find another Killing vector The metric is dependent on , so we cannot immediately deduce that it is a Killing vector. In fact, explicitly evaluating Killing's equation reveals it is not a Killing field. Intuitively, the flow generated by moves points downwards. Near , points move apart, thus distorting the metric, and we can see it is not an isometry, and therefore not a Killing field. Killing fields have the property that the Lie bracket of two Killing fields is still a Killing field. Thus, the Killing fields on a manifold M form a Lie subalgebra of vector fields on M. For example, we may wish to compute the dimension of this algebra and its structure constants, and possibly identify the Lie algebra. Since are a coordinate basis, their commutator vanishes. Unfortunately, as earlier mentioned, is not a Killing field. In these coordinates, it is difficult to guess other Killing fields. We can realize the algebra of Killing fields by considering as an embedded manifold of , and work in orthonormal Cartesian coordinates where commutators will turn out to be straight-forward. The generator is recognized as a rotation about the -axis A second generator, rotations about the -axis, is Commuting these two, one finds a third generator for rotations about the -axis The algebra given by linear combinations of these three generators closes, and we have the relations We identify this as the Lie algebra Expressing and in terms of spherical coordinates gives and We'd like to say these three Killing fields are a complete set of generators for the algebra. One way to see this is that as a submanifold of , its symmetries are a strict subset of those of . The symmetries of are generated by rotations and translations. This can be shown using Killing's equation. Translations do not preserve the unit sphere, but rotations do. Hence this is the complete set of generators of the algebra of Killing fields, which we can identify with the algebra of rotations. Killing fields in Minkowski space The Killing