Whether to capitalise/capitalize the name of an empire is a matter of style. British publications are inconsistent (or uneasy) about capitalising. The (UK) Guardian (centre-left) style guide gives 'British empire' as the expected style, whereas the Daily Telegraph (centre-right) guide has no mention of the term, but has plenty of articles with "British Empire" in them. The (American) Chicago Manual of Style is lukewarm and says that a writer risks...
...confusion by tampering with a fairly strong tradition of capping
officially recognized territories subject to an emperor or a king or
Many American academic and commercial style guides recommend capitalising, and the general rule seems to be to capitalise words derived from proper names. Webster's II New Riverside University Dictionary says:
Words derived from proper names, except in their extended senses: the
Whichever style you choose to follow, be consistent, that is, if you capitalise 'British Empire', and you mention others, you should do the same for them, for example the Roman, Ottoman, Byzantine, Holy Roman Empires. Also to note: you would never capitalise a zone of influence which is sometimes called an 'empire' of a nation, but is not officially ruled by that nation, so that if you write about the 'American empire', you should lower-case 'empire'.