I would like to know how to correctly combine the sentence 1. with the sentence 2.
- People are interested in how these algorithms work.
- I am writing a book about algorithms... [here I need to recast the above sentence so it fits]
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You would normally use an adverbial clause:
You could also do it in various other ways, of course; the possibilities are endless. But why not make two separate sentences? There is no inherent benefit in a single sentence. In fact, some sentences can be overloaded with information, which makes it harder to see what the author meant.
If I absolutely had to combine these exact sentence with a relative construction, I could only do it like this:
I find this construction very cumbersome and I do not recommend it, but it might occur in sloppy speech. You just can't properly use the subject of an embedded sentence as a relative pronoun like this in English, which is what you'd have to do if you wanted to integrate these two sentences word by word. I'd rather recast it:
Here I have substituted workings. But it sounds awkward: you just wouldn't combine these two sentences with a relative pronoun in natural language; and workings has somewhat different connotations.
"I am writing a book about algorithms because people are interested in how algorithms work."