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Theory and Modern Applications

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

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

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\(X_{1}\)

\(X_{2}\)

\(X_{3}\)

\(X_{4}\)

\(X_{5}^{f}\)

\(X_{6}^{g}\)

\(X_{7}^{h}\)

\(X_{8}^{d}\)

\(X_{1}\)

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