The product of two quaternions p = [ a b c d ] = a+bi+cj+dk and q = [ e f g h ] = e+fi+gj+hk is given by the quaternion

pq

= (a+bi+cj+dk)(e+fi+gj+hk)

= (ae-bf-cg-dh) + (af+be+ch-dg)i + (ag-bh+ce+df)j + (ah+bg-cf+de)k

= [ (ae-bf-cg-dh) (af+be+ch-dg) (ag-bh+ce+df) (ah+bg-cf+de) ].

To derive this formula, just remember the rules

i² = j² = k² = -1

ij = -ji = k

jk = -kj = i

ki = -ik = j,

distribute the multiplication, and collect terms.