The manager （主语）likes （谓语动词）to be democratic （宾语）by involving everyone in the decision making.（修饰不定式的方式状语）

我把意思分解一下：

1. The manager likes to be democratic

经理喜欢民主

2. by involving everyone in the decision making.

通过让每个人都参与决策的方式。

3. to be democratic：不定式作 likes 的宾语。

4. by involving everyone in the decision making：实现这个不定式行为的方式。