Inner Products, Group, Ring of Quaternion Numbers

The articles [9], [1], [3], [4], [6], [5], [2], [7], and [8] provide the notation and terminology for this paper. We use the following convention: q, r, c, c1, c2, c3 are quaternion numbers and x1, x2, x3, x4, y1, y2, y3, y4 are elements of R. 0H is an element of H. 1H is an element of H. Next we state several propositions: (1) For all real numbers x, y, z, w holds 〈x, y, z, w〉H = x+y · i+ z · j+w ·k. (2) (c1 + c2) + c3 = c1 + (c2 + c3). (3) c+ 0H = c. (4) −〈x1, x2, x3, x4〉H = 〈−x1,−x2,−x3,−x4〉H. (5) 〈x1, x2, x3, x4〉H − 〈y1, y2, y3, y4〉H = 〈x1 − y1, x2 − y2, x3 − y3, x4 − y4〉H. (6) (c1 − c2) + c3 = (c1 + c3)− c2. (7) c1 = (c1 + c2)− c2. (8) c1 = (c1 − c2) + c2. (9) (−x1) · c = −x1 · c.