(<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 ,1,0 *] 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 ,1,0 *] by QUATERN2:4 ;
hence <j> ^3 = - <j> ; :: thesis: verum