Would the Prophet (SAWW) and the Ahlulbayt (AS) be considered socialist in the modern sense even with owning fadak?

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The Islamic system of government cannot be accurately described by any of these forms or theories of government we see today. Here is a small booklet about the government of Islam

http://iswf.us/wp-content/uploads/2019/03/the-islamic-system-of-Goverment.pdf