Let Γ denote a bipartite distance-regular graph with vertex set X , diameter D ≥ 4 , and valency k ≥ 3 . Let C X denote the vector space over C consisting of column vectors with entries in C and rows indexed by X . For z ∈ X , let z ^ denote the vector in C X with a 1 in the z -coordinate, and 0 in all other coordinates. Fix a vertex x of Γ and let T = T ( x ) denote the corresponding Terwilliger algebra. Assume that up to isomorphism there exist exactly two irreducible T -modules with endpoint 2, and they both are thin. Fix y ∈ X such that ∂ ( x , y ) = 2 , where ∂ denotes path-length distance. For 0 ≤ i , j ≤ D define w i j = ∑ z ^ , where the sum is over all z ∈ X such that ∂ ( x , z ) = i and ∂ ( y , z ) = j . We define W = span { w i j ∣ 0 ≤ i , j ≤ D } . In this paper we consider the space M W = span { m w ∣ m ∈ M , w ∈ W } , where M is the Bose–Mesner algebra of Γ . We observe that MW is the minimal A -invariant subspace of C X which contains W , where A is the adjacency matrix of Γ . We show that 4 D - 6 ≤ dim ( M W ) ≤ 4 D - 2 . We display a basis for MW for each of these five cases, and we give the action of A on these bases.