Poincaré group is semi-product of translations and Lorentz-group

Hi everybody,

Can somebody help me with the following proof.
Show that for the Poincaré group

[itex]P=T\odot L[/itex]

Where [itex]T[/itex] is the group of translations and [itex]L[/itex] is the Lorentz group and [itex]P[/itex] is the semi-direct product of the two subgroups

I know the axioms for a semi-direct product in this case are

• [itex]T[/itex] is an invariant subgroup of [itex]P[/itex] while [itex]L[/itex] is any subgroup of [itex]P[/itex]
• [itex]T\cap L=\{E\}[/itex], where [itex]E[/itex] is the identity of [itex]P[/itex]
• For every element [itex]S[/itex] in [itex]P[/itex] we have an element [itex]{{S}_{1}}[/itex] of [itex]T[/itex] and an element [itex]{{S}_{2}}[/itex] of [itex]L[/itex] for which: [itex]S={{S}_{1}}\circ {{S}_{2}}[/itex]

How do I proof these axioms?