Australia, which is a federation of the states and territories, is officially the Commonwealth of Australia. Commonwealth is often used to refer to national institutions such as the government and the parliament. However, because of the potential for confusion with the Commonwealth of Nations (the British Empire), Australian should generally be used in place of Commonwealth. References to legislation are the main context requiring use of Commonwealth, which should always have an initial capital:

the Australian Government
the Australian Parliament
the Commonwealth Environment Protection and Biodiversity Conservation Act 1990
or
the Environment Protection and Biodiversity Conservation Act 1990 (Cwlth)