4. Do you delegate high visibility tasks?
Managers who don't do this work hard to hog all the glory, spend all their time in meetings with other managers or presenting to higher level people etc. It isn't good for them nor their reports, by not growing the reports skills they have a hard time getting to the next level as a manager. Also they create a situation where they are the bus factor 1, which is problematic in itself.
Feeling someone cares fits the warmth dimension; commitment to excellence is a hallmark of competence. I would say the word trust can encompass both.
[0] - https://en.m.wikipedia.org/wiki/Stereotype_content_model
(Unless you value getting screwed over, I guess ;)
And for a leader, you can scale these by caring about roles and what info those roles get to show trust.
Really cool framework as a fundamental baseline.
Everyone asks three questions of people leading them:
1. Can I trust you?
2. Do you care about me?
3. Are you committed to excellence? (aka do you have high standards).
Think about someone you admire. Odds are the answer to the above for them is "yes" to all three.
Now think of someone you have had a lot of problems with. Odds are the answers to the above are "no".
[0] - https://en.wikipedia.org/wiki/Lou_Holtz