The parameters 
,
,
,
and 
which, like the three Euler angles, provide a way
to uniquely characterize the orientation of a solid body. These parameters satisfy
the identities
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
and
 |  |  |
(6)
|
 |  |  |
(7)
|
where 
denotes the complex conjugate. In terms of the
Euler angles 
, 
, and 
, the Cayley-Klein parameters are given by
 |  |  |
(8)
|
 |  |  |
(9)
|
 |  |  |
(10)
|
 |  |  |
(11)
|
(Goldstein 1980, p. 155).
The transformation matrix is given in terms of the Cayley-Klein parameters by
![A=[1/2(alpha^2-gamma^2+delta^2-beta^2) 1/2i(gamma^2-alpha^2+delta^2-beta^2) gammadelta-alphabeta; 1/2i(alpha^2+gamma^2-beta^2-delta^2) 1/2(alpha^2+gamma^2+beta^2+delta^2) -i(alphabeta+gammadelta); betadelta-alphagamma i(alphagamma+betadelta) alphadelta+betagamma]](/images/equations/Cayley-KleinParameters/NumberedEquation1.svg) |
(12)
|
(Goldstein 1980, p. 153).
The Cayley-Klein parameters may be viewed as parameters of a matrix (denoted 
for its close relationship with quaternions)
![Q=[alpha beta; gamma delta]](/images/equations/Cayley-KleinParameters/NumberedEquation2.svg) |
(13)
|
which characterizes the transformations
 |  |  |
(14)
|
 |  |  |
(15)
|
of a linear space having complex axes. This matrix satisfies
 |
(16)
|
where 
is the identity matrix and 
the conjugate transpose,
as well as
 |
(17)
|
In terms of the Euler parameters 
and the Pauli matrices 
,
the 
-matrix
can be written as
 |
(18)
|
(Goldstein 1980, p. 156).
Matrices. Cayley-Klein Parameters." §1.4.3
in