TITLE: Matrix representation of the Quaternions?
QUESTION [16 upvotes]: Can anyone explain how why the matrix representation of the quaternions using real matrices is constructed as such?

REPLY [4 votes]: It is a simple consequence of the multiplication rules of the quaternions.   
So
$$
\eqalign{
  & \left( {a + bi + cj + dk} \right) \cdot 1 = \left( {\matrix{
   a & b & c & d  \cr 
   { - b} & a & { - d} & c  \cr 
   { - c} & d & a & { - b}  \cr 
   { - d} & { - c} & b & a  \cr 
 } } \right)^{\,T} \left( {\matrix{
   1  \cr 
   0  \cr 
   0  \cr 
   0  \cr 
 } } \right) = \left( {\matrix{
   1  \cr 
   0  \cr 
   0  \cr 
   0  \cr 
 } } \right)^{\,T} \left( {\matrix{
   a & b & c & d  \cr 
   { - b} & a & { - d} & c  \cr 
   { - c} & d & a & { - b}  \cr 
   { - d} & { - c} & b & a  \cr 
 } } \right) = \left( {\matrix{
   a  \cr 
   b  \cr 
   c  \cr 
   d  \cr 
 } } \right)  \cr 
  & \left( {a + bi + cj + dk} \right) \cdot i = \left( {\matrix{
   a & b & c & d  \cr 
   { - b} & a & { - d} & c  \cr 
   { - c} & d & a & { - b}  \cr 
   { - d} & { - c} & b & a  \cr 
 } } \right)^{\,T} \left( {\matrix{
   0  \cr 
   1  \cr 
   0  \cr 
   0  \cr 
 } } \right) = \left( {\matrix{
   { - b}  \cr 
   a  \cr 
   { - d}  \cr 
   c  \cr 
 } } \right) \cr} 
$$
and so on