Three new summation formulas for 6ψ6 bilateral basic hypergeometric series attached to root systems are presented. Remarkably, two of these formulae, labelled by the A2n−1 and A2n root systems, can be reduced to multiple 6φ5 sums generalising the well-known van Diejen sum. This latter sum serves as the weightfunction normalisation for the BCn q-Racah polynomials of van Diejen and Stokman. Two 8φ7-level extensions of the multiple 6φ5 sums, as well as their elliptic analogues, are conjectured. This opens up the prospect of constructing novel A-type extensions of the Koornwinder–Macdonald theory.