http://revach.net/article.php?id=3429

Section: Parshas HaShavua   Category: Quick Vort
Parshas Mishpatim: Meshech Chochma - Priestly Murder Of Last Resort
V'Chi Yazid Ish Al Rei'eihu LiHorgo B'Arma, Mei'im Mizbichi Tikachenu Lamus" But if a man plots against his neighbor to kill him intentionally, take him from My mizbei'ach to put him to death (Mishpatim 21:14).  What is a murderer doing by the mizbei'ach?

The Meshech Chochma answers that we find by the nations that often the cause of murder is to marry the wife of the murder victim.  This is what Avrohom was scared of with both Paroh and Avimelech.  Similarly the medrash tells us that the snake wanted to kill Adam so he can marry Chava and it uses the same word "Arma" to describe the snake.

Kohanim are prohibited from marrying divorced women but they marry a widow.  His only recourse, if he wants to marry a married woman, is to see the husband dead.  Therefore when a kohein, who is more likely to murder than someone else, murders someone. go take him away from the Mizbei'ach where he spends most of his time offering up the many korbanos that the women need to bring after childbirth and Zava.