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TableĀ 1 Commutator table

From: Lie symmetry analysis and invariant solutions of 3D Euler equations for axisymmetric, incompressible, and inviscid flow in the cylindrical coordinates

\([ \boldsymbol{V}_{\boldsymbol{1}}, \boldsymbol{V}_{\boldsymbol{2}}]\) \(\boldsymbol{X}_{\boldsymbol{1}}\) \(\boldsymbol{X}_{\boldsymbol{2}}\) \(\boldsymbol{X}_{\boldsymbol{3}}\) \(\boldsymbol{X}_{\boldsymbol{4}}\)
\(\boldsymbol{X}_{\boldsymbol{1}}\) 0 \(f_{3} ' \frac{\partial }{\partial z} + f_{3}^{\prime \prime } \frac{\partial }{\partial u} + a_{1} \frac{\partial }{\partial p}\) \(\frac{\partial }{\partial t} + a_{2} \frac{\partial }{\partial z} + a_{3} \frac{\partial }{\partial u} + a_{4} \frac{\partial }{\partial p}\) \(a_{5} \frac{\partial }{\partial z} + a_{6} \frac{\partial }{\partial u} + a_{7} \frac{\partial }{\partial p}\)
\(\boldsymbol{X}_{\boldsymbol{2}}\) \(-( f_{3} ' \frac{\partial }{\partial z} + f_{3}^{\prime \prime } \frac{\partial }{\partial u} + a_{1} \frac{\partial }{\partial p} )\) 0 \(-t f_{3} ' \frac{\partial }{\partial z} + a_{8} \frac{\partial }{\partial u} - \frac{2}{r^{2} v} \frac{\partial }{\partial v} + a_{9} \frac{\partial }{\partial p}\) \(f_{3} \frac{\partial }{\partial z} + f_{3} ' \frac{\partial }{\partial u} + \frac{4}{r^{2} v} \frac{\partial }{\partial v} + a_{10} \frac{\partial }{\partial p}\)
\(\boldsymbol{X}_{\boldsymbol{3}}\) \(-( \frac{\partial }{\partial t} + a_{2} \frac{\partial }{\partial z} + a_{3} \frac{\partial }{\partial u} + a_{4} \frac{\partial }{\partial p} )\) \(-(-t f_{3} ' \frac{\partial }{\partial z} + a_{8} \frac{\partial }{\partial u} - \frac{2}{r^{2} v} \frac{\partial }{\partial v} + a_{9} \frac{\partial }{\partial p} )\) 0 \(a_{11} \frac{\partial }{\partial z} + a_{12} \frac{\partial }{\partial u} + a_{13} \frac{\partial }{\partial p}\)
\(\boldsymbol{X}_{\boldsymbol{4}}\) \(-( a_{5} \frac{\partial }{\partial z} + a_{6} \frac{\partial }{\partial u} + a_{7} \frac{\partial }{\partial p} )\) \(-( f_{3} \frac{\partial }{\partial z} + f_{3} ' \frac{\partial }{\partial u} + \frac{4}{r^{2} v} \frac{\partial }{\partial v} + a_{10} \frac{\partial }{\partial p} )\) \(-( a_{11} \frac{\partial }{\partial z} + a_{12} \frac{\partial }{\partial u} + a_{13} \frac{\partial }{\partial p} )\) 0