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I am curious what is the exact meaning of the phrase "fast inside", as used in this sentence by D. H. Lawrence:

White savages, with motor-cars, telephones, incomes and ideals! Savages fast inside the machine; yet savage enough, ye gods!

Is "fast" actually an imperative verb?

In this sentence, fast is not a verb, it's an adverb, qualifying “inside the machine”. There is no verb in the main clause, just like there is no verb in the previous sentence. You can add “they are” at the beginning of each sentence if you want to make a full grammatical sentence.

It means “firmly attached” or more generally “difficult to remove”. The Cambridge English Dictionary (meaning C2) gives the definition “firmly fixed”. Merriam-Webster (adverb meaning 1) gives the definition “in a firm or fixed manner”. The adverb fast in this sense has the same etymology as the verb fasten. “Fastened” implies that there is a specific object (e.g. a rope, belt or bolt) whose purpose is to prevent movement. On the other hand, “fast” tends to mean that what prevents removal is some different physical process, or it can have a figurative meaning. An example of the concrete meaning (from CED) is:

The glue had set and my hand was stuck fast.

A common figurative meaning is the expression “fast asleep” (MW lists it as a separate meaning), which is a synonym of “sound asleep”: in a deep sleep, not easy woken up.

In Lawrence's sentence, the savages are “fast inside the machine”, meaning that they are in a world of machines and you can't take them apart from their machines. The use of the singular and the definite article for “the machine” (as opposed to “inside machines”) conveys the imagery that the world that the white savages inhabit is a sort of giant machine. It's not that they are in a car and you can't get them out of it, but that they live in a machine world.