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  Highest symmetric non-maximally symmetric spacetime

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What is the highest number of symmetries (Killing vectors) that a (4-dimensional) spacetime can have without being maximally symmetric? From what I can see, it seems to be 7 (which includes the Einstein universe and some pp-wave spacetimes), but the theorems used (in chapter 11 and 12 of Stephani's "Exact Solutions of the Einstein's Field Equations") only apply to spacetimes stemming from a variety of stress energy tensors (Vacuum, lambda vacuum, EM fields, perfect fluids, pure radiation).

Can that result be generalized to all spacetimes, regardless of their sources?

This post imported from StackExchange Physics at 2015-08-17 08:04 (UTC), posted by SE-user Slereah
asked Aug 16, 2015 in Theoretical Physics by Slereah (540 points) [ no revision ]
Please put the theorems here for those who do not have access to the text.

This post imported from StackExchange Physics at 2015-08-17 08:04 (UTC), posted by SE-user 0celo7
commented Aug 16, 2015 by 0celo7 (50 points) [ no revision ]

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