You're missing a step, which most adults also miss when dealing with adults.
You have to figure out why they lied, and not in an accusatory way. Often, particularly when someone wants us to promise something that we don't want to, we are trained to lie to get out of the situation. It can often feel like there is no alternative to lying.
This is, obviously, not an ideal behavior, but you can't solve it by force. You have to accept that they have a reason for not wanting to do what you want them to, and you have to come to a mutual understanding.
In any healthy interaction, you do not unilaterally set the rules. You both have input, and you have to respect theirs, even if you don't agree with it. That will help them learn to respect your input as well.
You have to figure out why they lied, and not in an accusatory way. Often, particularly when someone wants us to promise something that we don't want to, we are trained to lie to get out of the situation. It can often feel like there is no alternative to lying.
This is, obviously, not an ideal behavior, but you can't solve it by force. You have to accept that they have a reason for not wanting to do what you want them to, and you have to come to a mutual understanding.
In any healthy interaction, you do not unilaterally set the rules. You both have input, and you have to respect theirs, even if you don't agree with it. That will help them learn to respect your input as well.