Many of the identities on the home page suggest defining the septright triad, which comes in two varieties: one for rtv3s and one for qsu3s. Let a, b, c, d, e be integers, and:

• [rtv3 triad] Suppose there are three septrights of the following structure:

 X′ = ⟨ 0, ±a, 0, ±b, 0, ±c ⟩ Y′ = ⟨ 0, ±b, 0, ±c, 0, ±a ⟩ Z′ = ⟨ 0, ±c, 0, ±a, 0, ±b ⟩

The expression ± X′ ± Y′ ± Z′ represents 8 = 23 possible results. If at least one of those results is of the form

⟨ 0, +d, 0, −d, 0, +d

then X′, Y′, Z′ are defined to form an rtv3 triad.

• [qsu3 triad] Suppose there are three septrights of the following structure:

 X″ = ⟨ ±a, 0, ±b, 0, ±c, 0 ⟩ Y″ = ⟨ ±b, 0, ±c, 0, ±a, 0 ⟩ Z″ = ⟨ ±c, 0, ±a, 0, ±b, 0 ⟩

The expression ± X″ ± Y″ ± Z″ represents 8 possible results. If at least one of those results is of the form

⟨ +e, 0, +e, 0, −e, 0 ⟩

then X″, Y″, Z″ are defined to form a qsu3 triad.

If X, Y, Z form a triad, so do these:

• X, Y, Z
• X, −Y, Z
• X, Y, −Z

If most of the signs in the expression ± X ± Y ± Z are minus, it is canonical to negate all three of X, Y, Z.

add add add result QR · SV = + 2R − 2T + 3V SV · TU = + 2R − 3T + 2V TU · QR = + 3R − 2T + 2V + 7R − 7T + 7V
add add sub result 7R ÷ U = + 3Q + 5S + 1U 7T ÷ S = + 5Q − 1S − 3U 7V ÷ Q = + 1Q − 3S + 5U + 7Q + 7S − 7U
add add sub result QT · RS = + 4R − 1T + 2V RS · UV = + 1R − 2T + 4V UV · QT = − 2R + 4T − 1V + 7R − 7T + 7V
add add add result 7R ÷ Q = + 1Q + 4S − 2U 7T ÷ U = + 4Q + 2S − 1U 7V ÷ S = + 2Q + 1S − 4U + 7Q + 7S − 7U
 add add sub result Q ÷ U = 1 ÷ V = + R + V U ÷ S = 1 ÷ R = + R − T S ÷ Q = 1 ÷ T = + T − V + 2R − 2T + 2V
 add add add result Q ÷ R = + Q − U S ÷ V = + Q + S U ÷ T = + S − U + 2Q + 2S − 2U
 add add sub result T ÷ R = − 1R + 2T V ÷ T = + 1T − 2V R ÷ V = + 2R + 1V − 3R + 3T − 3V
add add sub result 7T ÷ Q = 7 ÷ S = − 1Q + 3S + 2U 7V ÷ U = 7 ÷ Q = + 3Q − 2S + 1U 7R ÷ S = 7 ÷ U = + 2Q + 1S + 3U zero
add add sub result R ÷ T = + 1R − 1T + 2V T ÷ V = + 2R − 1T + 1V V ÷ R = − 1R + 2T − 1V + 4R − 4T + 4V
add add sub result S ÷ T = + Q − S + U U ÷ R = − Q + S + U Q ÷ V = + Q + S + U − Q − S + U
 add add sub result A = + 1R − 5T + 5V B = + 5R − 1T + 5V C = − 5R + 5T − 1V + 11R − 11T + 11V
 add add sub result D = − 2R + 5T − 4V F = − 5R + 4T − 2V E = + 4R − 2T + 5V − 11R + 11T − 11V

Besides the sum of the elements of a triad, something might be said about their product.

Consider an rtv3 triad in the following form:

 X′ = ⟨ 0, +a, 0, −b, 0, +c ⟩ Y′ = ⟨ 0, +b, 0, −c, 0, +a ⟩ Z′ = ⟨ 0, +c, 0, −a, 0, +b ⟩

Clearly, X′ + Y′ + Z′ = a + b + c. Further, X′Y′Z′ = d where d is a ten-term cubic polynomial:

 d = abc − a3 − b3 − c3 − 3a2b − 3b2c − 3c2a + 4a2c + 4b2a + 4c2b

Now consider a qsu3 triad in the following form:

 X″ = ⟨ +a, 0, −b, 0, +c, 0 ⟩ Y″ = ⟨ +b, 0, −c, 0, +a, 0 ⟩ Z″ = ⟨ +c, 0, −a, 0, +b, 0 ⟩

Of course, X″ + Y″ + Z″ = (a + b + c)√7. Also, X″Y″Z″ = e√7 where e is:

 e = abc − a3 − b3 − c3 + 2a2b + 2b2c + 2c2a + a2c + b2a + c2b