View source for Quaternary quadratic form

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A [[Quadratic form|quadratic form]] in four variables. A quaternary quadratic form over a field  $  F $
is related to the algebra of quaternions (cf. [[Quaternion|Quaternion]]) over the same field. Namely, corresponding to the algebra with basis  $  [ 1 , i _ {1} , i _ {2} , i _ {3} ] $, 
$  i _ {1}  ^ {2} = - a _ {1} \in F $, 
$  i _ {2}  ^ {2} = - a _ {2} \in F $, 
and  $  i _ {1} i _ {2} = - i _ {2} i _ {1} = i _ {3} $, 
is the quaternary quadratic form which is the norm of the quaternion,

$$ 
q ( x _ {0} , x _ {1} , x _ {2} , x _ {3} )  = \ 
N ( x _ {0} + x _ {1} i _ {1} + x _ {2} i _ {2} + x _ {3} i _ {3} ) =
$$

$$ 
= \ 
x _ {0}  ^ {2} + a _ {1} x _ {1}  ^ {2} + a _ {2} x _ {2}  ^ {2} + a _ {1} a _ {2} x _ {3}  ^ {2} .
$$

For quaternary quadratic forms corresponding to quaternion algebras, and only for these, composition of quaternary quadratic forms is defined:

$$ 
q ( x) q ( y)  =  q ( z) ,
$$

where the coordinates of the vector  $  z $
are bilinear forms in  $  x $
and  $  y $. 
Composition of this kind is possible only for quadratic forms in two, four and eight variables.

====Comments====
The last-mentioned result is known as Hurwitz's theorem; see [[Quadratic form|Quadratic form]].