Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 55 / 1 / Solution

Use the Lie bracket convention $[X,Y]=XY-YX$. If $E_{ij}$ denotes a matrix unit, then $E_{ij}{E}_{kl}={\delta }_{jk}{E}_{il}$. Here this givesThus, writing $[X_j,X_k]=f^i{}_{jk}{X}_i$, the nonzero structure constants of a Lie algebra areAll other entries vanish. This is the Lie algebra of the translation-dilation group of the plane: its two-dimensional translation ideal is abelian, and $X_0$ acts on that ideal as the identity. In group coordinates the multiplication and inverse areThese formulas also make the semidirect product structure explicit.