Stand on your box of truth always. Lead by a good example of telling the truth and calling out all lies.

Unless it is something detrimental to you or others, I would completely ignore, people like that thrive on attention, even if it is negative

I feel ya on the inveterate liars. My boss is one, she lied on Friday and now she couldn't remember her lie. She forgets her lies. Lol. I would just ignore the dude.

Give him enough rope to hang himself.

The ego is a powerful drug. It causes people to do and say things that feed the beast as it eats away at respect. I allow liars to weave their web. But unless I am required to work with them, I avoid them openly. Fellow employees who are liars....I like to volunteer them for work that feeds their ego.

If his lies are important to the business you are in I would look for support from your own management. If not, I would just silently smile and roll my eyes when the liar is bloviating.

If you have an employment manual that provides instructions on what to do about this situation, follow the instructions in the manual. If there is no manual or the manual doesn’t cover this situation, discuss the matter with your supervisor.

I worked in critical rotating equipment. I always double checked our work. Ive found loose fasteners and missing ones and always brought it up to those that worked on it. Its challenging to punish someone that is trying to do a good job. I have not found a person being negligent. Im retired now...whew! Im glad too.

Depends on the person and the lie

Call him out if you have proof(it's good for you and maybe good for him)

This seems to really bother you. I can relate; I once had a boss like that and ended up quitting my job over it. I found out half the department followed suit within the next several weeks. But anyway, I think you need to ask yourself if it's worth it to you to get yourself upset about it. What I did was to do some research about something she bragged about and the next time she brought it up, asked her questions she couldn't answer, but remember, there may be a price to pay for upsetting him. You never know how someone like that will respond, and you don't want to put your job in jeopardy. Remember, you said he lies.

I say "I'm a pineapple" and walk away. They think I'm weird and have no idea what I'm talking about but if they want to make ridiculous claims so can I.

The truth, always. Integrity is everything.
I have lost jobs for that reason, but my conscience was clear.