I recently realized that there are roughly two kinds of people, namely people who are good at symbolic/abstract ideas and people who are good at real/physical/touchable things, which can be considered orthogonal to the extrovert/introvert categorization. For a
software
engineering team, I find it tends to be suboptimal to have a physical-thinking oriented manager who dislikes abstract, symbolic notions, even though they are brilliant and would probably a good fit for say electrical or mechanical engineering teams.
My sampling space isn't large enough, but it's interesting to keep an eye on the personalities of the managers in your organization from that perspective.