6 Quaternions and Octonions 49points of a plane. The definition of quaternions adopted here will be analogous to our
definition of complex numbers.
We d e fi n e aquaternionto be a 2×2 matrix of the formA=
(
ab
−b ̄ a ̄)
,
whereaandbare complex numbers and the bar denotes complex conjugation. The set
of all quaternions will be denoted byH. We may define addition and multiplication in
Hto be matrix addition and multiplication, sinceHis closed under these operations.
FurthermoreHcontains0 =
(
00
00
)
, 1 =
(
10
01
)
,
andA∈Himplies−A∈H.
It follows from the properties of matrix addition and multiplication that addition
and multiplication of quaternions have the properties(A2)–(A5)and(M3)–(M4), with
0 and 1 as identity elements for addition and multiplication respectively. However,
(M2)no longer holds, since multiplication isnot always commutative. For example,
(
01
− 10)(
0 i
i 0)
=
(
0 i
i 0)(
01
− 10
)
.
On the other hand, there are now two distributive laws:for all A,B,C∈H,A(B+C)=AB+AC,(B+C)A=BA+CA.It is easily seen thatA∈His in thecentreofH,i.e.AB=BAfor everyB∈H,
if and only ifA=λ 1 for some real numberλ. Since the mapλ→λ 1 preserves sums
and products, we can regardRas contained inHby identifying the real numberλwith
the quaternionλ 1.
We define theconjugateof the quaternion
A=
(
ab
−b ̄ a ̄)
,
to be the quaternion
A ̄=
(
a ̄ −b
ba ̄)
.
It is easily verified that
A+B=A ̄+B ̄, AB=B ̄A ̄, A ̄ ̄=A.Furthermore,AA ̄ =AA ̄=n(A), A+A ̄=t(A),