I believe that three things contribute to a white soap. First off, use light olive oil, not the green or dark yellow kind. Make sure the other oils you use are light in color and find FOs or EOs that do not discolor (mostly trial and error with the FOs but any vanilla in the fragrance will discolor). Also, do not gel. The soap will be lighter than the same soap that has gelled. It is true that lard can make for a white soap if the other ingredients do not discolor. The whitest soap I ever made was with palm oil that I bought at the health food store. Not an economical way to go, though. I also used coconut oil from the health food store, very light olive oil and lavender EO. I did gel that soap and it was blazing white. You can also lighten up a soap with a tiny bit of TD.