I am accustomed to the fact that he lies.

[Vladv1]

I know that some adjectives are not followed by -that clauses straight away, for example we can't say "I am accustomed that he lies", we need to say "I am accustomed to the fact that he lies" in order to place a that clause. But some adjectives are never followed by -that clauses, for example we can't say "We are devoted that/to the fact that". How do I know whether an adjective can be followed by -that clauses, or by the fact that, and when an adjectvie can't be followed by -that clause at all? Does it have to do with the nature of the adjective? I am really confused.

 

[emsr2d2]

I know that some adjectives are not immediately followed by -that clauses. straight away,

Note my changes above.

For example, we can't say "I am accustomed that he lies", (don't strike through words just because you've said we can't say it!) but we need to can/should say "I am accustomed to the fact that he lies" in order to place include a -that clause.

Note my changes above. Don't try to put "for example" in the middle of a sentence. Start a sentence with it and always follow it with a comma.
Bear in mind that native speakers are more likely to say "I am used to his/him lying". We don't use "accustomed to" anywhere near as often as learners seem to think.

But However, some adjectives are never followed by -that clauses. For example, we can't say "We are devoted that/to the fact that ...".

As a learner, don't try to start a sentence with "but". Use it as a conjunction. If you're tempted to put it at the start, change it to "However" and follow it with a comma.
The reason no one would say "We are devoted to the fact that ..." is that we don't devote ourselves to facts!

How do I know whether an adjective can be followed by a -that clauses clause no comma here or "by the fact that", and when an adjective can't be followed by a -that clause at all? Does it have to do with the nature of the adjective? I am really confused.

Note my changes above.

I'm sure a grammar expert will come up with some kind of rule (if there is one) but after a little thought, I can only come up with the idea that it's mostly adjectives that describe some kind of feeling/emotion that can take "that", but not all of them.

I'm happy that you're here.
I'm confident that I can complete the task.
I'm worried that he won't make it in time.
I'm scared that I'll fall.
I'm delighted that you're coming to my party.

I'm tired that ...
I'm exhausted that ...
I'm used to that ...
I'm bored that ...

 

[jutfrank]

I know that some adjectives are not followed by -that clauses straight away, for example we can't say "I am accustomed that he lies", we need to say "I am accustomed to the fact that he lies" in order to place a that clause.

Right.

But some adjectives are never followed by -that clauses, for example we can't say "We are devoted that/to the fact that".

This is not right, and is probably the origin of your confusion. After 'devoted' you can't use a 'that'-clause compliment but you can use a 'to' preposition phrase. So, if you want to connect 'devoted to' to a 'that'-clause, then you need to use the 'fact' as a grammatical connector. The issue, however, is whether it makes sense to do that, and it quite probably doesn't make sense to do that.

How do I know whether an adjective can be followed by -that clauses

Chiefly, as usual, you know from the meaning of the adjective, and whether it makes sense to combine it with the informational content of the 'that'-clause.

Does it have to do with the nature of the adjective?

Yes. The complement pattern has to do with the meaning. We use 'that'-clauses to express some kind of propositional content, or in other words to state something true about the world. Example:

My wife is having an affair.

This is a proposition. That means that it says something true about a state-of-affairs of the world. This kind of propositional content works very well with certain verbs.

I know that my wfe is having an affair.

The verb 'know' makes good sense here, because of its meaning, which is related to the idea of truth, knowledge and certainty.

I'm sure that my wife is having an affair.

The adjective 'sure' also makes good sense here, because the propositional content of the 'that'-clause is something that it is possible to be sure about.

Words such as 'know' and 'sure' say something about knowledge and certainty, but words that express an emotional response also work very well with the propositional content of 'that'-clauses:

I'm actually glad that my wife is having an affair.

 

[jutfrank]

I'm sure a grammar expert will come up with some kind of rule (if there is one) but after a little thought, I can only come up with the idea that it's mostly adjectives that describe some kind of feeling/emotion that can take "that", but not all of them.

Yes. I didn't read your post until after I wrote mine, but you're pretty much halfway there. It isn't just 'feeling' adjectives, but equally words that express a degree of knowledge or certainty in some way.

I'm happy that you're here.
I'm confident that I can complete the task.
I'm worried that he won't make it in time.
I'm scared that I'll fall.
I'm delighted that you're coming to my party.

blue = feeling
red = level of certainty