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Table 1 The commutator table of \({\mathfrak{g}}\)

From: Conservation laws and exact solutions of the \((3+1)\)-dimensional Jimbo–Miwa equation

[,] \(X_{1}\) \(X_{2}\) \(X_{3}\) \(X_{4}\) \(X_{5}^{f}\) \(X_{6}^{g}\) \(X_{7}^{h}\) \(X_{8}^{d}\)
\(X_{1}\) 0 \(-3X_{2}\) 0 0 \(2X_{5}^{f}\) \(X_{6}^{3t g'-g}\) \(-X_{7}^{h}\) \(X_{8}^{3t d_{t}+d}\)
\(X_{2}\) \(3X_{2}\) 0 0 0 \(X_{7}^{3f'/4}\) \(X_{6}^{g'}\) 0 \(X_{8}^{d_{t}}\)
\(X_{3}\) 0 0 0 \(-X_{4}\) \(X_{5}^{zf'-f}\) 0 \(X_{7}^{zh'}\) \(X_{8}^{zd_{z}}\)
\(X_{4}\) 0 0 \(X_{4}\) 0 \(X_{5}^{f'}\) 0 \(X_{7}^{h'}\) \(X_{8}^{d_{z}}\)
\(X_{5}^{\tilde{f}}\) \(-2X_{5}^{\tilde{f}}\) \(-X_{7}^{3\tilde{f}'/4}\) \(-X_{5}^{z\tilde{f}'-\tilde{f}}\) \(-X_{5}^{\tilde{f}'}\) \(X_{5}^{3t(f\tilde{f}''-\tilde{f}f'')/4}\) \(X_{8}\) \(X_{8}\) 0
\(X_{6}^{\tilde{g}}\) \(-X_{6}^{3t \tilde{g}'-\tilde{g}}\) \(-X_{6}^{\tilde{g}'}\) 0 0 \(-X_{8}\) \(X_{8}\) \(X_{8}\) 0
\(X_{7}^{\tilde{h}}\) \(X_{7}^{\tilde{h}}\) 0 \(-X_{7}^{z\tilde{h}'}\) \(-X_{7}^{\tilde{h}'}\) \(-X_{8}\) \(-X_{8}\) 0 0
\(X_{8}^{\tilde{d}}\) \(-X_{8}^{3t \tilde{d}_{t}+\tilde{d}}\) \(-X_{8}^{\tilde{d}_{t}}\) \(-X_{8}^{z\tilde{d}_{z}}\) \(-X_{8}^{\tilde{d}_{z}}\) 0 0 0 0