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or
N(
q
1
q
2
).
The algebra of octonions also includes this
composition law
.
In the algebra of reals
N(
q
1
)N(
q
2
)
=
,wehave
x
1
y
1
=
(x
1
y
1
)
2
.
In the algebra of complex numbers
R
C
,wehave
z
1
=
x
1
+
x
2
i
z
2
=
y
1
+
y
2
i
x
1
+
x
2
y
1
+
y
2
=
x
2
y
2
)
2
x
2
y
1
)
2
.
(x
1
y
1
−
+
(x
1
y
2
+
In the algebra of quaternions
H
,wehave
q
1
=
x
1
+
x
2
i
+
x
3
j
+
x
4
k
q
2
=
y
1
+
y
2
i
+
y
3
j
+
y
4
k
x
1
+
x
4
y
1
+
y
4
=
x
2
+
x
3
+
y
2
+
y
3
+
x
4
y
4
)
2
+
(x
1
y
2
+
x
2
y
1
+
x
3
y
4
−
x
4
y
3
)
2
+
(x
1
y
3
−
x
2
y
4
+
x
3
y
1
+
x
4
y
2
)
2
+
(x
1
y
4
+
x
2
y
3
−
x
3
y
2
+
x
4
y
1
)
2
.
(x
1
y
1
−
x
2
y
2
−
x
3
y
3
−
And in the algebra of octonions
O
, we have something similar, but with many more
terms.
The above algebras are called composition algebras because of their inherent
composition law, and Adolf Hurwitz proved that such algebras can only exist in 1,
2, 4 and 8 dimensions.
