expression
under the orders of; by order of
Fixed formal expression used when an action is taken on someone's command or authority. Often appears in official or historical contexts.
彼は社長の命により、新しいプロジェクトを始めた。
He started a new project under the orders of the company president.
その兵士は上官の命により行動した。
The soldier acted by order of his superior officer.
命令 is a general noun for 'order' or 'command', while 命により is a formal adverbial phrase meaning 'under the orders of'.
指示 is a neutral noun for 'instruction' or 'direction', less authoritative than 命により, which implies a stronger command.
Combination of 命 (order, command) + により (formal expression meaning 'by' or 'according to'). The phrase is a fixed formal construction.