Alligator lizard eggs (at least Southern Alligator) are fairly easy to hatch.

You do not need to provide any heat above room temperature. You can, but if you do, try not to go above 82F.

Humid substrate such as vermiculite or coconut fiber. Should be moist but not wet. I make small depressions so the eggs do not accidentally roll, as that kills them.

Use a plastic tub with saran wrap kept on via rubberband to keep the humidity up.

Last year the method that worked for me:

Potato Salad Tub with Coconut Fiber - I filled it 3/4 full because I figured the high volume would help stabalize temperature as I did not use a thermostat.

The top inch of fiber was dryer than the rest - I had the lower intentionally really wet so that I wouldn't have to add water. It worked. The top inch was moist but not wet.

I kept them at about 80F (but not thermostat controlled) and they hatched in about 45 days. But you can keep them at room temperature and it just takes a little longer.
-----
3.6 L. getula californiae
1.1 L. getula nigrita
1.0 Boa constrictor constrictor (suriname, fostering/rescue)
2.3 Elgaria multicarinata multicarinata