On a seperate level, why is the death knight a lich? Liches are usually arcane creatures, and the power's name "Spellmaster" also makes it look like it's meant for arcane abilities. Death knights are usually soldiers and defenders with martial prowess.
Perhaps another template than Lich would fit better, and avoid issues of recharging via spellmaster?
The simplest answer would have to be I wanted to make a truly frightening undead solo monster, and those two templates seem to blend together to create a truly terrifying one. Upon looking at the Lich template, I can't really find anything that states it must use its powers on abilities with the Arcane power source - indeed, I can't find anything that contradicts a Fighter becoming a lich, so long as their intelligence and level are high enough. Thematically the lich also fits very well, making the character both an excellent melee fighter and someone who it is dangerous to draw too close to.
Storywise, it's an enemy who was cursed by the Gods to never die, based upon the Russian folklore figure Koschei the Deathless. A contemporary of Baba Yaga, he serves as the prototype for the lich, because he hid away his soul inside of a needle, inside of an egg, inside of a duck, inside of a fish, inside of a hare, inside of an iron chest buried at the foot of a tree on a distant island and could not be killed until the needle was destroyed.
So thematically and powers wise Lich is a smooth fit, but I am uncertain of how this power interacts with the death knight's recharging ability, or if I should stick with recharging base abilities that are supposed to be encounter only. Theoretically those should be much stronger and more battle changing, but at the same time, the possibility of spamming Unholy Flames each round is kinda scary, particularly with minions. The How to Design an Encounter thread makes me feel a bit safer though, since the encounter is being built at party level +2.
