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