All translations are unitary and abelian. Translations can be written in terms of unit vectors τ τ = ∑ ∑ i = 1 3 n i a i We can think of these as commuting operators τ τ ^ ^ = τ τ ^ ^ 1 τ τ ^ ^ 2 τ τ ^ ^ 3 where τ τ ^ ^ i = n i a ^ ^ i