Word choice between 推舉 , 推選 , 薦舉
In 「推舉」,「推」means &quot;to push &quot;, and 「舉」 means &quot;to raise &quot;

「推舉」means to &quot;single out and elevate someone&quot; which means &quot; to recommend&quot;

In 「薦舉」,「薦」means &quot; to recommend&quot;

「薦舉」means &quot; to elevate and recommend someone&quot;

In「推選」, 「選」means &quot; to choose&quot;

「推選」means &quot;to single out and choose someone&quot; / &quot; to nominate and elect someone&quot;

Examples :

我推舉他為經理 ( I recommend him to be the manager)
我薦舉他為經理 ( I recommend him to be the manager)
眾人推舉之下，他成為了經理 ( After everybody recommended him, he became the manager)
眾人薦舉之下，他成為了經理 ( After everybody recommended him, he became the manager)
眾人推選他為經理 ( Everybody recommend and choose him to be the manager)

*「推舉」;「薦舉」;「推選」only apply to people
*「推薦」is the most common term for &quot;to recommend&quot;, followed by「推舉」and then「薦舉」
*「推薦」apply to either people or object
Example:
向日本人推薦美國牛肉 ( recommend American beef to Japanese)
我推薦他為經理 ( I recommend him to be the manager)
*「薦舉」is more commonly written as 「舉薦」
*「選舉」(election)