Should 'to be' be involved or it's optional?

Question

Context:

"The nervous system was so complex and highly developed as to leave Lake aghast. Though excessively primitive and archaic in some respects, the thing had a set of ganglial centres and connectives arguing the very extremes of specialised development. Its five-lobed brain was surprisingly advanced; and there were signs of a sensory equipment, served in part through the wiry cilia of the head, involving factors alien to any other terrestrial organism."

(Source)

I think that 'to be' verb should be involved:

Factors ARE alien to any other terrestrial organism

• Oxford's elaboration of the definition (2.Unfamiliar and disturbing or distasteful)

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Questions:

1. Why didn't the author use 'to be' verb in the sentence?

2. If it's optional- could you please explain these options to me?

Answer 1

It's an example of a reduced defining relative clause.

• ...and there were signs of a sensory equipment, served in part through the wiry cilia of the head, involving factors alien to any other terrestrial organism.

A reduced relative clause is a relative clause that is not marked by an explicit relative pronoun or complementizer such as who, which or that.

From ThoughtCo:

Relative clauses can also be reduced to shorter forms if the relative clause modifies the subject of a sentence. Relative clause reduction refers to removing a relative pronoun to reduce.

Here the subject is "factors", And the relative clause was reduced to a prepositional phrase:

• ...involving factors that (which) are alien to any other terrestrial organism.

To reduce a relative clause to a prepositional phrase we need to remove the relative pronoun - (that/which in your example), remove the verb 'be' - (are in your example), and place the prepositional phrase after the modified noun - ('factors' in your example).

Here are more examples of defining relative clause reduced to prepositional phrases:

• The box which was on the table was made in Italy. > Reduced: The box on the table was made in Italy.
• The woman who was at the meeting spoke about business in Europe. > Reduced: The woman at the meeting spoke about business in Europe.

Answer 2

It's a participle adjective clause or reduced defining relative clause. Let's have a look on your sentence:

"there were signs of a sensory equipment, served in part through the wiry cilia of the head, involving factors alien to any other terrestrial organism."

In this interesting sentence you see both adverbial participles and participle adjectives. In details, the adverbial participles are

• "served in part through the wiry cilia of the head"

• "involving factors alien to any other terrestrial organism"

and the adjective participle is

• alien to any other terrestrial organism

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Now let's get back to your questions.

1. Why didn't the author use 'to be' verb in the sentence?

Briefly, they didn't do so because it's wrong. However, the author could say:

Factors which(that) are alien to any other terrestrial organism.

If you do some research on "defining relative clause", you will find that the sentence above can be reduced to

Factors alien to any other terrestrial organism.

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2. If it's optional- could you please explain these options to me?

Regarding why they preferred to use the adjective participle or reduced relative clause, I would say the reduced version is more eloquent and compact which is preferable, especially in academic or formal texts.