The phrase "keep reviewing" is correct and usable in written English. It is often used as encouragement or advice, for example, "Remember to keep reviewing the material to make sure you understand!".
He might want to keep reviewing those videos.
It's important that you keep reviewing your brand at regular intervals.
Or maybe the justices want to spare other courts the need to keep reviewing and declaring unconstitutional an endless assortment of violent-video bans.
Unlike hedgehogs, foxes also had a desire to keep reviewing their assumptions, updating their estimates, and improving their understanding — a kind of attitude that Tetlock calls "perpetual beta".
In the event, the FSA kept reviewing the board structure of the bank until October 2006 when it concluded that the "risks associated with CEO dominance and challenge to him had been mitigated sufficiently that the issue could be closed".
I kept reviewing all the mistakes I made: I should have trained harder and more consistently.
However, ER, LM and OM kept reviewing the data being collected daily together with the research assistants.
Being a terminologist, I care about word choice. Ludwig simply helps me pick the best words for any translation. Five stars!
Maria Pia Montoro
Terminologist and Q/A Analyst @ Translation Centre for the Bodies of the European Union