Now I lie me down to sleep.
The 1850-ish Pinneo’s Grammar which I used for homeschooling my kids is athwart your and others’ contentions.
I lay down, lay an egg, and lie about it.
I lay down, laid an egg,and lied about it.
I have lain down, laid an egg, and lied about it.
Maybe a more modern adaptation of the language?
I don’t believe it’s a modern grammar rule. Use lay when there’s a direct object, lie when the object is itself.
lay, laid, laid, laying
lie, lay, lain, lying
He surrendered by laying the gun down.
He surrendered by lying down.