Usage patterns
〈人〉と意気が合うto click/hit it off with someone
We clicked with each other from the moment we first met.
意気を示すto show one's spirit/resolve
The players showed their fighting spirit before the match.
意気を挫くto dampen someone's spirits/morale
The harsh criticism dampened his spirits.
Expressions
意気が揚がるbe in high spirits; be elated; be buoyant 意気に感じるbe moved by someone's spirit; be inspired by someone's mettle 意気に燃えるbe fired up with enthusiasm; burn with zeal 意気天を衝くbe in high spirits; be full of fighting spirit