Discussion of family F−.
Section 4−. Each of the algebras within family F− has a different definition of multiplication, but within each the following properties must always be satisfied.
Multiplication is anticommutative:
d × e = − e × d
Multiplication is not associative, but it does satisfy the flexible identity:
a × (b × a) = (a × b) × a
Multiplication also satisfies the Jabobi identity:
d × (e × f) + e × (f × d) + f × (d × e) = 0
Multiplication distributes over addition on both sides:
(d + e) × f = d × f + e × f
f × (d + e) = f × d + f × e
Zero is absorbent: 0 × d = 0.
Because of anticommutativity, there cannot be an identity for multiplication.
Multiplicative rotativity is like that of the F+ family:
d^{⊐} × e^{⊐} = (d × e)^{⊐}
The product of two atoms is not necessarily an atom, but will be a linear combination of atoms.
With addition and multiplication as defined here, each algebra of family F− is a Lie algebra.
Finally, (d × q)_{a} is not required to equal d_{a} × q_{a}.
Section 5−. Because multiplication distributes over addition, multiplication in all the algebras in F− can be defined by these nine atomic products:
a × a  b × a  c × a 
a × b  b × b  c × b 
a × c  b × c  c × c 
By anticommutativity, a × a = b × b = c × c = 0.
Let t, u, v be integer constants. If a × b = ⟨ t, u, v ⟩ then by rotativity b × c = ⟨ v, t, u ⟩ and c × a = ⟨ u, v, t ⟩.
By anticommutativity, b × a = − ⟨ t, u, v ⟩, c × b = − ⟨ v, t, u ⟩ and a × c = − ⟨ u, v, t ⟩.
To summarize:
a × a =  ⟨ 0, 0, 0 ⟩  b × a =  − ⟨ t, u, v ⟩  c × a =  + ⟨ u, v, t ⟩  
a × b =  + ⟨ t, u, v ⟩  b × b =  ⟨ 0, 0, 0 ⟩  c × b =  − ⟨ v, t, u ⟩  
a × c =  − ⟨ u, v, t ⟩  b × c =  + ⟨ v, t, u ⟩  c × c =  ⟨ 0, 0, 0 ⟩ 
Incidentally,
(a × c)^{⊐} = − a × b
(b × a)^{⊐} = − b × c
(c × b)^{⊐} = − c × a
A script in the F− case can reduce to an ordered quadruple, with a semicolon after the initial zero:
⟨ 0; t, u, v ⟩
Equivalently in full:
⟨ 0; (a × b)_{a}, (a × b)_{b}, (a × b)_{c} ⟩
Equivalently in brief:
⟨ 0; a × b ⟩
Section 6−. The simplest nontrivial multiplications in family F− are the one shown immediately below, and its negative. The one shown is named SM− (for simple multiplication), and its script is ⟨ 0; 0, 0, +1 ⟩:
a × a  = ⟨  0,  0,  0 ⟩  b × a  = ⟨  0,  0,  −1 ⟩  c × a  = ⟨  0,  +1,  0 ⟩  
a × b  = ⟨  0,  0,  +1 ⟩  b × b  = ⟨  0,  0,  0 ⟩  c × b  = ⟨  −1,  0,  0 ⟩  
a × c  = ⟨  0,  −1,  0 ⟩  b × c  = ⟨  +1,  0,  0 ⟩  c × c  = ⟨  0,  0,  0 ⟩ 
More compactly:
a × a = b × b = c × c = 0
a × b = c = −b × a
b × c = a = −c × b
c × a = b = −a × c
The script of SM−'s negative simply inverts all the signs: ⟨ 0; 0, 0, +1 ⟩.
Applying a constant factor to all the products yields a valid but different multiplication — and consequently a different algebra. The same effect can also be achieved by applying the constant factor to each term of the script. The following example is the quintuple of SM−:
a × a  = ⟨  0,  0,  0 ⟩  b × a  = ⟨  0,  0,  −5 ⟩  c × a  = ⟨  0,  +5,  0 ⟩  
a × b  = ⟨  0,  0,  +5 ⟩  b × b  = ⟨  0,  0,  0 ⟩  c × b  = ⟨  −5,  0,  0 ⟩  
a × c  = ⟨  0,  −5,  0 ⟩  b × c  = ⟨  +5,  0,  0 ⟩  c × c  = ⟨  0,  0,  0 ⟩ 
If that constant factor happens to be zero, the result becomes trivial. Two trivial F− multiplications.
Here are four examples of nontrivial multiplications:
Script ⟨ 0; +1, 0, −1 ⟩:
a × a  = ⟨  0,  0,  0 ⟩  b × a  = ⟨  −1,  0,  +1 ⟩  c × a  = ⟨  0,  −1,  +1 ⟩  
a × b  = ⟨  +1,  0,  −1 ⟩  b × b  = ⟨  0,  0,  0 ⟩  c × b  = ⟨  +1,  −1,  0 ⟩  
a × c  = ⟨  0,  +1,  −1 ⟩  b × c  = ⟨  −1,  +1,  0 ⟩  c × c  = ⟨  0,  0,  0 ⟩ 
Script ⟨ 0; +1, −6, +5 ⟩:
a × a  = ⟨  0,  0,  0 ⟩  b × a  = ⟨  −1,  +6,  −5 ⟩  c × a  = ⟨  −6,  +5,  +1 ⟩  
a × b  = ⟨  +1,  −6,  +5 ⟩  b × b  = ⟨  0,  0,  0 ⟩  c × b  = ⟨  −5,  −1,  +6 ⟩  
a × c  = ⟨  +6,  −5,  −1 ⟩  b × c  = ⟨  +5,  +1,  −6 ⟩  c × c  = ⟨  0,  0,  0 ⟩ 
Script ⟨ 0; +2, −5, +3 ⟩:
a × a  = ⟨  0,  0,  0 ⟩  b × a  = ⟨  −2,  +5,  −3 ⟩  c × a  = ⟨  −5,  +3,  +2 ⟩  
a × b  = ⟨  +2,  −5,  +3 ⟩  b × b  = ⟨  0,  0,  0 ⟩  c × b  = ⟨  −3,  −2,  +5 ⟩  
a × c  = ⟨  +5,  −3,  −2 ⟩  b × c  = ⟨  +3,  +2,  −5 ⟩  c × c  = ⟨  0,  0,  0 ⟩ 
Script ⟨ 0; +5, +2, −7 ⟩:
a × a  = ⟨  0,  0,  0 ⟩  b × a  = ⟨  −5,  −2,  +7 ⟩  c × a  = ⟨  +2,  −7,  +5 ⟩  
a × b  = ⟨  +5,  +2,  −7 ⟩  b × b  = ⟨  0,  0,  0 ⟩  c × b  = ⟨  +7,  −5,  −2 ⟩  
a × c  = ⟨  −2,  +7,  −5 ⟩  b × c  = ⟨  −7,  +5,  +2 ⟩  c × c  = ⟨  0,  0,  0 ⟩ 
Section 7−. Provided as an extended set of examples, the scripts below represent all the valid multiplications within family F− meeting the following four criteria:


