(<j> * <j>) * <j> = [*((((0 * 0) - (0 * 0)) - (1 * 1)) - (0 * 0)),((((0 * 0) + (0 * 0)) + (0 * 0)) - (0 * 1)),((((0 * 1) + (0 * 1)) + (0 * 0)) - (0 * 0)),((((0 * 0) + (0 * 0)) + (0 * 1)) - (1 * 0))*] * [*0,0,jj,0*] by QUATERNI:def 10
.= [*(((((- 1) * 0) - (0 * 0)) - (0 * 1)) - (0 * 0)),(((((- 1) * 0) + (0 * 0)) + (0 * 0)) - (0 * 1)),(((((- 1) * 1) + (0 * 0)) + (0 * 0)) - (0 * 0)),(((((- 1) * 0) + (0 * 0)) + (0 * 1)) - (0 * 0))*] by QUATERNI:def 10
.= - [*0,0,jj,0*] by QUATERN2:4 ;
hence <j> ^3 = - <j> ; :: thesis: verum