 This notation allows us to write the product of two dual quaternions as

The addition of dual quaternions is defined componentwise so that given, 

and
then

Multiplication of two dual quaternion follows from the multiplication rules for the quaternion units i, j, k and commutative multiplication by the dual unit ε.In particular, given

and
then

Notice that there is no "BD" term, because the definition of dual numbers requires that .This gives us the multiplication table (note the multiplication order is row times column)