carve, engrave
栔 centers on cutting into a surface to create a lasting mark: carving or engraving.
栔 combines 㓞 (a component associated with cutting or engraving) with 木 (tree, wood), suggesting the act of carving into wood.
The top part 㓞 looks like a tool for cutting, and the bottom 木 is wood. Picture a sharp tool carving a design into a wooden block.
For ケイ, imagine carving a wooden key (ケイ) as a gift: key -> ケイ, and the carved key fits perfectly.