The switching element for a Leonard pair

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Let V denote a vector space with finite positive dimension. We consider a pair of linear transformations A: V → V and A*: V → V that satisfy (i) and (ii) below: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A* is diagonal. (ii) There exists a basis for V with respect to which the matrix representing A* is irreducible tridiagonal and the matrix representing A is diagonal. We call such a pair a Leonard pair on V. Let v₀,v₁,...,vd (resp. w₀,w₁,...,wd) denote a basis for V referred to in (i) (resp. (ii)). We show that there exists a unique linear transformation S: V → V that sends v₀ to a scalar multiple of vd, fixes w₀, and sends wᵢ to a scalar multiple of wᵢ for 1 ≤ i ≤ d. We call S the switching element. We describe S from many points of view.