Here, the elements of the Weyl group are considered as orthogonal transformations, acting by matrix multiplication, on the real vector space of roots. If − I is an element of the Weyl group of a semisimple Lie algebra, then w 0 = − I. In the case of s l ( 2 , C ) , the Weyl group is W = { I, − I }. It follows that each π μ, μ = 0, 1,... is isomorphic to its dual π π μ μ ∗ ∗ . The root system of s l ( 2 , C ) ⊕ ⊕ s l ( 2 , C ) is shown in the figure to the right. The Weyl group is generated by { w γ γ } where w γ γ is reflection in the plane orthogonal to γ as γ ranges over all roots. Inspection shows that w α ⋅ w β = − I so − I ∈ W. Using the fact that if π, σ are Lie algebra representations and π ≅ σ, then Π ≅ Σ, the conclusion for SO(3; 1) is π π m , n ∗ ∗ ≅ ≅ π π m , n , Π Π m , n ∗ ∗ ≅ ≅ Π Π m , n , 2 m , 2 n ∈ ∈ N .