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Let us denote by COILS(v) a (3, 2, 1)-conjugate orthogonal idempotent Latin square of order v, and by ICOILS(v, n) an incomplete COILS(v) missing a sub-COILS(n). We shall investigate the existence of ICOILS(v, n). The construction of an ICOILS(8, 2) has already been instrumental in the construction of a COILS(26), the existence of Which was unknown for some time. A necessary condition for the existence of an. ICOILS(v, n) is v ~> 3n + 1. In this paper, it is shown that for all n ~> 1, andoi:10.1016/0012-365x(87)90207-x fatcat:5fmdqjyzs5eotpm2lm3ev7z4s4