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Theory and Modern Applications

Table 1 The multiplication table AB

From: \(\operatorname{Spin}(7)\)-structure equation and the vector elliptic Liouville equation

AB

n

\(e_{1}\)

\(e_{2}\)

\(e_{3}\)

\(\bar{e}_{1}\)

\(\bar{e}_{2}\)

\(\bar{e}_{3}\)

n

\(\sqrt{2}n \)

0

0

0

0

\(\sqrt{2}\bar{e}_{1}\)

\(\sqrt{2}\bar{e}_{2}\)

\(\sqrt{2}\bar{e}_{3}\)

\(e_{1}\)

\(\sqrt{2}e_{1} \)

0

\(-\sqrt{2}\bar{e}_{3}\)

\(\sqrt{2}\bar{e}_{2}\)

0

\(-\sqrt{2}\bar{n}\)

0

0

\(e_{2}\)

\(\sqrt{2}e_{2} \)

\(\sqrt{2}\bar{e}_{3}\)

0

\(-\sqrt{2}\bar{e}_{1}\)

0

0

\(-\sqrt{2}\bar{n}\)

0

\(e_{3}\)

\(\sqrt{2}e_{3} \)

\(-\sqrt{2}\bar{e}_{2}\)

\(\sqrt{2}\bar{e}_{1}\)

0

0

0

0

\(-\sqrt{2}\bar{n}\)

0

\(\sqrt{2}e_{1}\)

\(\sqrt{2}e_{2}\)

\(\sqrt{2}e_{3}\)

\(\sqrt{2}\bar{n} \)

0

0

0

\(\bar{e}_{1}\)

0

\(-\sqrt{2}n\)

0

0

\(\sqrt{2}\bar{e}_{1} \)

0

\(-\sqrt{2}e_{3}\)

\(\sqrt{2}e_{2}\)

\(\bar{e}_{2}\)

0

0

\(-\sqrt{2}n\)

0

\(\sqrt{2}\bar{e}_{2}\)

\(\sqrt{2}e_{3}\)

0

\(-\sqrt{2}e_{1}\)

\(\bar{e}_{3}\)

0

0

0

\(-\sqrt{2}n\)

\(\sqrt{2}\bar{e}_{3}\)

\(-\sqrt{2}e_{2}\)

\(\sqrt{2}e_{1}\)

0