DIE, DYE AND DAI

Dye is a chemical substance used to colour cloth. Cloth is dipped into a solution of dye by the dyer.
"Died in the wool" : we mean they are a person who holds fast to their opinions, beliefs or habits.
 If you died, you would be dead or deceased. Stop being alive.
"The die is cast" = "Crossed the Rubicon": means that you have embarked on a course of action that can´t be changed.
Dai is a Welsh name, the Welsh form of David.
A dai is a cube with 1 to 6 dots on each side, too.