The Redeemer of Blood and Why the Torah Specifies Wood

Or with a wooden implement, why is it stated? Because it says, "And if a man strikes his slave or his female slave with a rod," I understand that whether he struck him with a matter that can cause death or with a matter that cannot cause death, it is stated, "Or with a wooden implement," to indicate that he is not liable unless he struck him with a matter that can cause death. And from where do we derive that in a place that can cause death? It is derived from the phrase, "And if a man is quarrelsome, etc.," which indicates that he is not liable unless he struck him with a matter that can cause death. And concerning a place that can cause death, I have only that if he struck him with a wooden implement, he is liable. From where do we derive that throwing boards and wooden beams upon him are also included? It is stated, "He is a murderer; he shall surely be put to death." From where do we derive that if he struck him with a stone, you would say that he struck him with a clump of salt or with a dried fig cake, or he dropped a basket filled with dirt or a basket filled with pebbles upon him? It is derived from the phrase, "And if he struck him with a stone in his hand," or with a wooden implement. From where do we derive that if he struck him on the head of a beam or on the helm of a ship, you would say he is liable? It is stated, "Or with a wooden implement in his hand." From where do we derive that if one pushes his fellow into water or into fire, or incites a snake against him, you would say it is only if he is able to cause his death? It is stated, "In his hand," to include even if he cannot cause his death. As it is said, "The hand of the witnesses shall be first upon him," and if he fled to Babylon, they could go there to execute him. You would say that the redeemer of blood should kill him, etc. From where do we derive that even if he said, "I am unable to accept him," it is stated, "When he strikes him," to include even in a case where he cannot accept him. The redeemer of blood should kill the murderer; it is a mitzvah for the redeemer of blood to do so. And from where do we derive that if he does not have a redeemer of blood, the court appoints a redeemer of blood? It is stated, "When he strikes him." Why is it stated "redeemer of blood"? Because it says, "And if he struck him with an iron implement, or with a stone in his hand," [or with a wooden implement in his hand]. I have only that if he killed him with these, he is liable, but with what did he kill him?