Using the above identities (interpreted as pointwise addition of functions), for SO(3; 1) yields π π m , n ¯ ¯ = π π m , n + + π π m , n − − ¯ ¯ = π π m ⊕ ⊕ 2 n + 1 ¯ ¯ + π π n ¯ ¯ ⊕ ⊕ 2 m + 1 ¯ ¯ = π π n ⊕ ⊕ 2 m + 1 + π π m ¯ ¯ ⊕ ⊕ 2 n + 1 = π π n , m + + π π n , m − − = π π n , m 2 m , 2 n ∈ ∈ N Π Π m , n ¯ ¯ = Π Π n , m where the statement for the group representations follow from exp(X) = exp(X). It follows that the irreducible representations (m, n) have real matrix representatives if and only if m = n. Reducible representations on the form (m, n) ⊕ (n, m) have real matrices too.