Ibn Zyad himself said that Yazid ordered him to martyr Imam Hussain (May Allah be well pleased with him) on the threat of death. Imam Ibn Atheer (May Allah shower His Mercy on him) records the confession of Ibn Zyad.
Translation: Now, about my martyring Hussain (May Allah be well pleased with him), the fact is that Yazid ordered me to do this. He threatened to kill me if I didn't, so I martyred Imam Hussain (May Allah be well pleased with him).
According to Islamic Law, if any person murders anybody else, then as a punishment, the murderer is killed. But Yazid neither killed Ibn Zyad, Shimar or other officials nor dismissed them from their offices. Instead, he expressed his happiness. Afterwards, out of fear that the situation might go out of control he expressed some sorrow in a formal, political manner. In fact, this wretched man used his stick to poke at the teeth of Imam Hussain (May Allah be well pleased with him), those teeth, which the Holy Prophet (Sallallahu alaihi wa sallam) kissed.
Imam Ibn Atheer (May Allah shower His Mercy on him) has written in Tarikh Kamil, Imam Ibn Katheer (May Allah shower His Mercy on him) in Bidayah wan Nihayah, Imam Tabari (May Allah shower His Mercy on him) in Tarikh Tabari.
Translation: Abu Muhnaf relates on the authority of Abu Hamza Thimali, who relates on the authority of Abdullah Yamani, who relates on the authority of Qasim, who said: When the blessed head of Imam Hussain (May Allah be well pleased with him) was placed before Yazid, he had a stick in his hands with which he started poking at the teeth of Imam Hussain (May Allah be well pleased with him). Then he said: Verily, ours and his example is as said by Hisseen bin Hamam Mari: Our swords smash the skulls of those who have strength and domination over us; and of those who are really disobedient and oppressors.
Hadhrat Abu Bazrah Aslami (May Allah be well pleased with him) said: Listen O Yazid! By Allah! Your stick is touching that spot, which I witnessed the Holy Prophet (Sallallahu alaihi wa sallam) used to kiss. Then he said: Beware, O Yazid! On the day of judgment, Imam Hussain (May Allah be well pleased with him) will arrive with such grandeur that his intercessor will be the Holy Prophet (Sallallahu alaihi wa sallam) and you will arrive in such a state that Ibn Zyad will be yours."
In Bidayah wan Nihayah, Vol 8, Pg No. 215, at the end of the aforementioned Hadith, it is mentioned,
Translation: At that time, Hadhrat Abu Bazrah Aslami (May Allah be well pleased with him) said: Remove your stick! By Allah! I have often seen the Holy Prophet (Sallallahu alaihi wa sallam) place His mouth on the mouth of Imam Hussain (May Allah be well pleased with him) and kiss it.