Could the European Union (or an element of it) be described as a government?
The Wikipedia definition of government seems quite broad:
A government is the system or group of people governing an organized community, often a state.
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There is a wide variety of constitutional arrangements on Earth. Some have a written constitution, some do not. The EU has a number of attributes of a government, for example the EU parliament. EU documents talk about EU citizenship, and this citizenship gives many rights usually reserved for the citizens of a state. Only the EU is allowed to negotiate trade agreements for the member states.
A government is usually an attribute of a state, and of a sovereign state in particular. The EU is not sovereign by itself, it is an union of sovereign states which delegate the exercise of some their sovereign powers to the EU. EU member states are allowed to leave in accordance with their own constitutional requirements. Citizens of other member states may be deported from EU states.