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More can be said about the quadruples, diamond or otherwise. Let these be twelve arbitrary complex numbers:

 ap aq ar as bp bq br bs cp cq cr cs

Then let dp, dq, dr and ds be whatever values satisfy the following eight-dimensional vector equation using the baseline cross product:

 [ dp, dq, dr, ds, 0, 0, 0, 0 ] = [ ap, aq, ar, as, 0, 0, 0, 0 ] × [ bp, bq, br, bs, 0, 0, 0, 0 ] × [ cp, cq, cr, cs, 0, 0, 0, 0 ]

With those numbers, examples of closure and anti-closure can be given. The blue table below shows what closure looks like with the 14 diamond quadruples, and it corresponds fully to what was discussed on the home page. After that, the red table reveals the anti-closure (the meaning will become clear) obtained with the 56 non-diamond quadruples. Note that ±dp, ±dq, ±dr and ±ds are sufficient for all products.

in full
 ♦0123 if A = [ ap, aq, ar, as, 0, 0, 0, 0 ] if B = [ bp, bq, br, bs, 0, 0, 0, 0 ] if C = [ cp, cq, cr, cs, 0, 0, 0, 0 ] then A × B × C = [ +dp, +dq, +dr, +ds, 0, 0, 0, 0 ]
 ♦4567 if A = [ 0, 0, 0, 0, ap, aq, ar, as ] if B = [ 0, 0, 0, 0, bp, bq, br, bs ] if C = [ 0, 0, 0, 0, cp, cq, cr, cs ] then A × B × C = [ 0, 0, 0, 0, −dp, −dq, −dr, −ds ]
 ♦0145 if A = [ ap, aq, 0, 0, ar, as, 0, 0 ] if B = [ bp, bq, 0, 0, br, bs, 0, 0 ] if C = [ cp, cq, 0, 0, cr, cs, 0, 0 ] then A × B × C = [ +dp, +dq, 0, 0, +dr, +ds, 0, 0 ]
 ♦2367 if A = [ 0, 0, ap, aq, 0, 0, ar, as ] if B = [ 0, 0, bp, bq, 0, 0, br, bs ] if C = [ 0, 0, cp, cq, 0, 0, cr, cs ] then A × B × C = [ 0, 0, −dp, −dq, 0, 0, −dr, −ds ]
 ♦0167 if A = [ ap, aq, 0, 0, 0, 0, ar, as ] if B = [ bp, bq, 0, 0, 0, 0, br, bs ] if C = [ cp, cq, 0, 0, 0, 0, cr, cs ] then A × B × C = [ +dp, +dq, 0, 0, 0, 0, +dr, +ds ]
 ♦2345 if A = [ 0, 0, ap, aq, ar, as, 0, 0 ] if B = [ 0, 0, bp, bq, br, bs, 0, 0 ] if C = [ 0, 0, cp, cq, cr, cs, 0, 0 ] then A × B × C = [ 0, 0, −dp, −dq, −dr, −ds, 0, 0 ]
in brief  ♦0246 ABC = [ +dp, 0, +dq, 0, +dr, 0, +ds, 0 ] ♦1357 ABC = [ 0, −dp, 0, −dq, 0, −dr, 0, −ds ]
♦0257 ABC = [ −dp, 0, −dq, 0, 0, −dr, 0, −ds ] ♦1346 ABC = [ 0, +dp, 0, +dq, +dr, 0, +ds, 0 ]
♦0347 ABC = [ −dp, 0, 0, −dq, −dr, 0, 0, −ds ] ♦1256 ABC = [ 0, +dp, +dq, 0, 0, +dr, +ds, 0 ]
♦0356 ABC = [ −dp, 0, 0, −dq, 0, −dr, −ds, 0 ] ♦1247 ABC = [ 0, +dp, +dq, 0, +dr, 0, 0, +ds ]

in full
 0124 if A = [ ap, aq, ar, 0, as, 0, 0, 0 ] if B = [ bp, bq, br, 0, bs, 0, 0, 0 ] if C = [ cp, cq, cr, 0, cs, 0, 0, 0 ] then A × B × C = [ 0, 0, 0, +ds, 0, −dr, +dq, −dp ]
 3567 if A = [ 0, 0, 0, ap, 0, aq, ar, as ] if B = [ 0, 0, 0, bp, 0, bq, br, bs ] if C = [ 0, 0, 0, cp, 0, cq, cr, cs ] then A × B × C = [ +ds, −dr, +dq, 0, −dp, 0, 0, 0 ]
 0125 if A = [ ap, aq, ar, 0, 0, as, 0, 0 ] if B = [ bp, bq, br, 0, 0, bs, 0, 0 ] if C = [ cp, cq, cr, 0, 0, cs, 0, 0 ] then A × B × C = [ 0, 0, 0, +ds, +dr, 0, −dp, −dq ]
 3467 if A = [ 0, 0, 0, ap, aq, 0, ar, as ] if B = [ 0, 0, 0, bp, bq, 0, br, bs ] if C = [ 0, 0, 0, cp, cq, 0, cr, cs ] then A × B × C = [ −dr, −ds, +dq, 0, 0, +dp, 0, 0 ]
 0126 if A = [ ap, aq, ar, 0, 0, 0, as, 0 ] if B = [ bp, bq, br, 0, 0, 0, bs, 0 ] if C = [ cp, cq, cr, 0, 0, 0, cs, 0 ] then A × B × C = [ 0, 0, 0, +ds, −dq, +dp, 0, −dr ]
 3457 if A = [ 0, 0, 0, ap, aq, ar, 0, as ] if B = [ 0, 0, 0, bp, bq, br, 0, bs ] if C = [ 0, 0, 0, cp, cq, cr, 0, cs ] then A × B × C = [ −dr, +dq, +ds, 0, 0, 0, −dp, 0 ]
in brief  0127 ABC = [ 0, 0, 0, +ds, +dp, +dq, +dr, 0 ] 3456 ABC = [ +dq, +dr, +ds, 0, 0, 0, 0, +dp ]
0134 ABC = [ 0, 0, −ds, 0, 0, −dr, −dp, −dq ] 2567 ABC = [ −dr, −ds, 0, −dq, −dp, 0, 0, 0 ]
0135 ABC = [ 0, 0, −ds, 0, +dr, 0, −dq, +dp ] 2467 ABC = [ −ds, +dr, 0, −dq, 0, +dp, 0, 0 ]
0136 ABC = [ 0, 0, −ds, 0, +dp, +dq, 0, −dr ] 2457 ABC = [ +dq, +dr, 0, −ds, 0, 0, −dp, 0 ]
0137 ABC = [ 0, 0, −ds, 0, +dq, −dp, +dr, 0 ] 2456 ABC = [ +dr, −dq, 0, −ds, 0, 0, 0, +dp ]
and 40 others

Two diamond quadruples are complementary if they have no number in common, as ♦0257 and ♦1346.

Let DQ1 and DQ2 be two complementary diamond quadruples. If two factors of a baseline cross product conform to DQ1, and the other factor conforms to DQ2, then the product itself will also conform to DQ2. However, it is difficult to predict exactly what numbers will appear as components of the product.

Examples of the resulting patterns are given in the green table below. Two symbols are used: Z for zero, and N for a number that is not necessarily zero.

♦0123 if A = [N, N, N, N, Z, Z, Z, Z] if B = [N, N, N, N, Z, Z, Z, Z] if C = [Z, Z, Z, Z, N, N, N, N] then A × B × C = [Z, Z, Z, Z, N, N, N, N]
♦4567 if A = [Z, Z, Z, Z, N, N, N, N] if B = [Z, Z, Z, Z, N, N, N, N] if C = [N, N, N, N, Z, Z, Z, Z] then A × B × C = [N, N, N, N, Z, Z, Z, Z]
♦0145 if A = [N, N, Z, Z, N, N, Z, Z] if B = [N, N, Z, Z, N, N, Z, Z] if C = [Z, Z, N, N, Z, Z, N, N] then A × B × C = [Z, Z, N, N, Z, Z, N, N]
♦2367 if A = [Z, Z, N, N, Z, Z, N, N] if B = [Z, Z, N, N, Z, Z, N, N] if C = [N, N, Z, Z, N, N, Z, Z] then A × B × C = [N, N, Z, Z, N, N, Z, Z]
♦0167 if A = [N, N, Z, Z, Z, Z, N, N] if B = [N, N, Z, Z, Z, Z, N, N] if C = [Z, Z, N, N, N, N, Z, Z] then A × B × C = [Z, Z, N, N, N, N, Z, Z]
♦2345 if A = [Z, Z, N, N, N, N, Z, Z] if B = [Z, Z, N, N, N, N, Z, Z] if C = [N, N, Z, Z, Z, Z, N, N] then A × B × C = [N, N, Z, Z, Z, Z, N, N]
♦0246 if A = [N, Z, N, Z, N, Z, N, Z] if B = [N, Z, N, Z, N, Z, N, Z] if C = [Z, N, Z, N, Z, N, Z, N] then A × B × C = [Z, N, Z, N, Z, N, Z, N]
♦1357 if A = [Z, N, Z, N, Z, N, Z, N] if B = [Z, N, Z, N, Z, N, Z, N] if C = [N, Z, N, Z, N, Z, N, Z] then A × B × C = [N, Z, N, Z, N, Z, N, Z]
♦0257 if A = [N, Z, N, Z, Z, N, Z, N] if B = [N, Z, N, Z, Z, N, Z, N] if C = [Z, N, Z, N, N, Z, N, Z] then A × B × C = [Z, N, Z, N, N, Z, N, Z]
♦1346 if A = [Z, N, Z, N, N, Z, N, Z] if B = [Z, N, Z, N, N, Z, N, Z] if C = [N, Z, N, Z, Z, N, Z, N] then A × B × C = [N, Z, N, Z, Z, N, Z, N]
♦0347 if A = [N, Z, Z, N, N, Z, Z, N] if B = [N, Z, Z, N, N, Z, Z, N] if C = [Z, N, N, Z, Z, N, N, Z] then A × B × C = [Z, N, N, Z, Z, N, N, Z]
♦1256 if A = [Z, N, N, Z, Z, N, N, Z] if B = [Z, N, N, Z, Z, N, N, Z] if C = [N, Z, Z, N, N, Z, Z, N] then A × B × C = [N, Z, Z, N, N, Z, Z, N]
♦0356 if A = [N, Z, Z, N, Z, N, N, Z] if B = [N, Z, Z, N, Z, N, N, Z] if C = [Z, N, N, Z, N, Z, Z, N] then A × B × C = [Z, N, N, Z, N, Z, Z, N]
♦1247 if A = [Z, N, N, Z, N, Z, Z, N] if B = [Z, N, N, Z, N, Z, Z, N] if C = [N, Z, Z, N, Z, N, N, Z] then A × B × C = [N, Z, Z, N, Z, N, N, Z]