I've often wondered about people who appear to be very smart, and do very stupid things.  One theory is that people are smart and stupid independently in different domains.  Another theory is that "smart" and "stupid" are oversimplifications.  In line with the second theory, here is an ad-hoc set of axes of intelligence, based only on my own observations.

Memory

This may cover several sub-categories.  The advantages of a good memory should be obvious.

Ability to follow instructions

This is more important than it sounds.  I'm including the ability to verify mathematical proofs, and the ability to design a regression analysis for a psychology study.  Things that aren't research or engineering, but taking known solutions and applying them.

I came up with this because I have some friends who have difficult, complex jobs that they are good at - and yet frequently say incoherent things.

Doctors are smart.  Yet doctors are forbidden by law from being creative.  They're great at following instructions.  Becoming an MD means transforming yourself into a giant look-up table.  You memorize anatomy, physiology, diseases, symptoms, diagnostics, and treatments.  For every disease, there is an approved set of clinical diagnostic criteria, an approved set of laboratory diagnostics, an approved way of interpreting those tests, an approved way of presenting the results to the patient, and an approved set of possible treatments.  After using your knowledge to observe the patient, rule out some branches of the tree of possibilities, make further tests, and discern what the underlying problem is, if instead of retrieving the approved treatment from your look-up table, you ask the engineering question, "How can we fix this?", you are on the path to losing your license.

This ability is strongly correlated with memory.

Ability to think outside the box

This may be the same thing as creativity; but calling it "thinking outside the box" is less vague.  "Ability to not conform" is also part of it.  This may be anti-correlated with memory and the ability to follow instructions; it's probably more difficult to think of new approaches if old ones leap quickly to mind, just like it's difficult to compose music if every theme in your head turns into a Beatles tune after two bars.

This is the distinction between an M.S. and a Ph.D. (and between an M.D. and a Ph.D. - can you tell which I have?)  The only purpose of the years of agony of doing a dissertation no one will read is to show that you can do something original.

You can be great at thinking outside the box, and still be crazy.  Google Ment.if.ex, without the dots.  (Do not write his name in the comments without the dots.  Writing his name online summons him.  I'm not joking.)

Ability to notice success and failure

A friend kept telling me about a woman he knew who he thought would be great for me.  He told me she was smart, pretty, friendly, and fun.  Eventually I gave in and told him I'd like to meet her.

Instead of doing what had worked on me - telling her that I was right for her, smart, pretty, etc. - he told her that I was interested in her.  Predictably, she said, "That's creepy - I don't even know him!"  I asked him how it was that, in his thirty-some years of life, he hadn't noticed that that never works.  He said he wanted to be straightforward, not sneaky.

Sadly, morals are a big cause of not being able to notice success and failure.  Someone who believes they're doing the right thing doesn't allow themself to ask whether they succeed or fail.

The friend I mentioned is very good at following instructions.  If you're following instructions, you might not be checking up on whether you're succeeding or failing.  Rationalists are often bad at noticing success and failure.  Maybe it's because we're good at following instructions - instructions on how to be rational.  We're likely to follow our program of, say, trying to reason someone into a political view, or into liking us, without noticing that that doesn't work.

Categorization

This is a big one.  It's pretty close to "analytical ability".  By categorization I mean the ability to notice when two words mean the same thing, or similar things, or different things.  Or when two situations or systems are similar.  Or when one assumption is really two or three.  Analogical reasoning requires good categorization skills.  So does analytic thinking.  A major fault in most people's analytic ability is their inability to keep their terms straight, and use them consistently.

I include under categorization, the ability to generalize appropriately for the task.  Overgeneralizing during analysis leads to sloppy thinking; undergeneralizing while brainstorming stifles creativity.

Social intelligence

Is this a useful primitive category?  Lots of people think it is.  Perhaps I don't have enough of it to understand it.