You can lay an egg, lay something down, lay concrete or slabs, even lay a lover, but you yourself lie down in the present, and only lay down in the past.
That egg may be new-laid, your patio paving may be well-laid and your lover be well and truly laid. Both of you may even need to lie down for a while to recover. You won't need to lay down, unless you lay down the law.
You are lying down now, you lay down earlier, you have lain around for hours. If you have laid, it is the egg, the paving, or the lover. Time to crow?