Given a Lie group algebra with generators $ X_{i} $ how can I calculate the Casimir generator $ H= X_{i}X^{i} $? If possible with two examples please
The generator of translation in 2 dimensions $ P_{i} $ i=x,y with commutation relations $[P_{i},P_{j} ]=0 $
The generator of the angular momentum with commutation relations $ [L_{i} , L_{j}]=\epsilon _{ijk}L_{k} $
Here $ X^{i} $ is the 'dual' of $ X_{i} $.