Theory and Modern Applications
From: \(\operatorname{Spin}(7)\)-structure equation and the vector elliptic Liouville equation
A∖B | n | \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | n̄ | \(\bar{e}_{1}\) | \(\bar{e}_{2}\) | \(\bar{e}_{3}\) |
---|---|---|---|---|---|---|---|---|
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 | \(-\sqrt{2}\bar{n}\) |
n̄ | 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 |