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 d_p, d_q, d_r and d_s 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 ±d_p, ±d_q, ±d_r and ±d_s are sufficient for all products.

	diamond quadruples
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 ]

	non-diamond quadruples
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.

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