AbstractThree elements U, V, and W in the complete lattice I(A) of weak⁎-closed inner ideals in a JBW⁎-triple A are said to be in general position when (U,V) form a rigidly collinear pair, and (U,W) and (V,W) form orthogonal pairs. A complete description of the supremum U∨V∨W in I(A) in terms of the Peirce spaces corresponding to U, V, and W is given for the case in which all three inner ideals are Peirce and three other conditions are satisfied. By considering an example in the Albert JBW⁎-triple H3(O) it is shown that none of these is redundant and, therefore, that the result obtained is the best possible.