The racah algebra as a subalgebra of the bannai–ito Algebra

Assume that F is a field with char F ≠ 2. The Racah algebra ℜ is a unital associative F-algebra defined by generators and relations. The generators are A, B, C, D and the relations assert that [A, B] = [B, C] = [C, A] = 2D and each of [A, D] + AC − BA, [B, D] + BA − CB, [C, D] + CB − AC is central in ℜ. The Bannai–Ito algebra BI is a unital associative F-algebra generated by X, Y, Z and the relations assert that each of {X, Y } − Z, {Y, Z} − X, {Z, X} − Y is central in BI. It was discovered that there exists an F-algebra homomorphism ζ: ℜ → BI that sends A ↦→^{(2X−3)(2X+1)}₁₆, B ↦→^(2Y−3)(2Y₁₆⁺¹⁾, C ↦→^{(2Z−3)(2Z+1)}₁₆ . We show that ζ is injective and therefore ℜ can be considered as an F-subalgebra of BI. Moreover we show that any Casimir element of ℜ can be uniquely expressed as a polynomial in {X, Y } − Z, {Y, Z} − X, {Z, X} − Y and X + Y + Z with coefficients in F.