While giving a basis in a 3-dimensional space is a valid definition for a spherical tensor, it only covers the case for when the rank k is 1. For higher ranks, one may use either the commutator, or rotation definition of a spherical tensor. The commutator definition is given below, any operator T q ( k ) that satisfies the following relations is a spherical tensor: [ J ± ± , T q ( k ) ] = ℏ ℏ ( k ∓ ∓ q ) ( k ± ± q + 1 ) T q ± ± 1 ( k ) [ J z , T q ( k ) ] = ℏ ℏ q T q ( k )