\begin{definition}[Definition:Quaternion/Multiplication]
The '''product''' of two quaternions $\mathbf x_1 = a_1 \mathbf 1 + b_1 \mathbf i + c_1 \mathbf j + d_1 \mathbf k$ and $\mathbf x_2 = a_2 \mathbf 1 + b_2 \mathbf i + c_2 \mathbf j + d_2 \mathbf k$ is defined as:
{{begin-eqn}}     
{{eqn | l = \mathbf x_1 \mathbf x_2
      | o = :=
      | r = \paren {a_1 a_2 - b_1 b_2 - c_1 c_2 - d_1 d_2} \mathbf 1
      | c = 
}}
{{eqn | o = 
      | ro= +
      | r = \paren {a_1 b_2 + b_1 a_2 + c_1 d_2 - d_1 c_2} \mathbf i
      | c = 
}}
{{eqn | o = 
      | ro= +
      | r = \paren {a_1 c_2 - b_1 d_2 + c_1 a_2 + d_1 b_2} \mathbf j
      | c = 
}}
{{eqn | o = 
      | ro= +
      | r = \paren {a_1 d_2 + b_1 c_2 - c_1 b_2 + d_1 a_2} \mathbf k
      | c = 
}}
{{end-eqn}}
\end{definition}