The proper spaces of the matrix $NN'$, where $N$ is the incidence matrix of a triangular partially balanced incomplete block design, are exhibited explicitly; this provides a convenient form for the Gramian of a basis of the join of two of these spaces.
"Proper Spaces Related to Triangular Partially Balanced Incomplete Block Designs." Ann. Math. Statist. 31 (2) 498 - 501, June, 1960. https://doi.org/10.1214/aoms/1177705916