Jihad means striving. It is used to describe the inner struggle against sin as well as the outer struggle against those who would oppose Islam. Did Muhammad teach Holy War upon those who reject Islam? Yes he did. Following are various quotes from the Hadith dealing with killing people.
. The second best deed is to participate in Jihad
. Allah's Apostle was asked, "What is the best deed?" He replied, "To believe in Allah and His Apostle (Muhammad). The questioner then asked, "What is the next (in goodness)? He replied, "To participate in Jihad (religious fighting) in Allah's Cause." The questioner again asked, "What is the next (in goodness)?" He replied, "To perform Hajj (Pilgrim age to Mecca) 'Mubrur, (which is accepted by Allah and is performed with the intention of seeking Allah's pleasure only and not to show off and without committing a sin and in accordance with the traditions of the Prophet)." Volume 1, Book 2, Number 25, Narrated Abu Huraira:
. Muhammad said if someone leaves Islam, to kill him
. Ali burnt some people and this news reached Ibn 'Abbas, who said, "Had I been in his place I would not have burnt them, as the Prophet said, 'Don't punish (anybody) with Allah's Punishment.' No doubt, I would have killed them, for the Prophet said, 'If somebody (a Muslim) discards his religion, kill him.'" Volume 4, Book 52, Number 260 , Narrated Ikrima. Also, see Volume 9, Book 84, Number 64 , Narrated 'Ali.
. Muhammad approves of killing someone who hurt him and having the killer lie

. "Allah's Apostle said, "Who is willing to kill Ka'b bin Al-Ashraf who has hurt Allah and His Apostle?" Thereupon Muhammad bin Maslama got up saying, "O Allah's Apostle! Would you like that I kill him?" The Prophet said, "Yes," Muhammad bin Maslama said, "Then allow me to say a (false) thing (i.e. to deceive Kab). "The Prophet said, "You may say it." Then Muhammad bin Maslama went to Kab and said, "That man (i.e. Muhammad demands Sadaqa (i.e. Zakat) from us, and he has troubled us, and I have come to borrow something from you." On that, Kab said, "By Allah, you will get tired of him!" Muhammad bin Maslama said, "Now as we have followed him, we do not want to leave him unless and until we see how his end is going to be. .." Volume 5, Book 59, Number 369 , Narrated Jabir bin 'Abdullah.
. Paradise was guaranteed to the first to take part in a naval battle
. That 'Umair bin Al-Aswad Al-Anasi told him that he went to 'Ubada bin As-Samit while he was staying in his house at the sea-shore of Him with (his wife) Um Haram. 'Umair said. Um Haram informed us that she heard the Prophet saying, "Paradise is granted to the first batch of my followers who will undertake a naval expedition." Um Haram added, I said, 'O Allah's Apostle! Will I be amongst them?' He replied, 'You are amongst them.' The Prophet then said, 'The first army amongst' my followers who will invade Caesar's City will be forgiven their sins.' I asked, 'Will I be one of them, O Allah's Apostle?' He replied in the negative." Volume 4, Book 52, Number 175 Narrated Khalid bin Madan:
. Those who fight in Jihad have the right to the spoils of the conquered or Paradise if he dies
. The Prophet said, "The person who participates in (Holy battles) in Allah's cause and nothing compels him to do so except belief in Allah and His Apostles, will be recompensed by Allah either with a reward, or booty (if he survives) or will be admitted to Paradise (if he is killed in the battle as a martyr). Had I not found it difficult for my followers, then I would not remain behind any sariya going for Jihad and I would have loved to be martyred in Allah's cause and then made alive, and then martyred and then made alive, and then again martyred in His cause." Volume 1, Book 2, Number 35 , Narrated Abu Huraira.
. ". . . "This is the Will of Allah, "After the people returned, the Prophet sat and said, "Anyone who has killed an enemy and has a proof of that, will posses his spoils." I got up and said, "Who will be a witness for me?" and then sat down. The Prophet again said, "Anyone who has killed an enemy and has proof of that, will possess his spoils." I (again) got up and said, "Who will be a witness for me?" and sat down. Then the Prophet said the same for the third time. I again got up, and Allah's Apostle said, "O Abu Qatada! What is your story?" Then I narrated the whole story to him. A man (got up and) said, "O Allah's Apostle! He is speaking the truth, and the spoils of the killed man are with me. So please compensate him on my behalf." On that Abu Bakr As-Siddiq said, "No, by Allah, he (i.e. Allah's Apostle ) will not agree to give you the spoils gained by one of Allah's Lions who fights on the behalf of Allah and His Apostle." The Prophet said, "Abu Bakr has spoken the truth." So, Allah's Apostle gave the spoils to me. I sold that armor (i.e. the spoils) and with its price I bought a garden at Bani Salima, and this was my first property which I gained after my conversion to Islam." Volume 4, Book 53, Number 370 , Narrated Abu Qatada.
. ". . . . While we were in this state, the Lord of the Heavens and the Earths, Elevated is His Remembrance and Majestic is His Highness, sent to us from among ourselves a Prophet whose father and mother are known to us. Our Prophet, the Messenger of our Lord, has ordered us to fight you till you worship Allah Alone or give Jizya (i.e. tribute); and our Prophet has informed us that our Lord says:-- Whoever amongst us is killed (i.e. martyred), shall go to Paradise to lead such a luxurious life as he has never seen, and whoever amongst us remain alive, shall become your master." (Al-Mughira, then blamed An-Numan for delaying the attack and) An-Nu' man said to Al-Mughira, "If you had participated in a similar battle, in the company of Allah's Apostle he would not have blamed you for waiting, nor would he have disgraced you. But I accompanied Allah's Apostle in many battles and it was his custom that if he did not fight early by daytime, he would wait till the wind had started blowing and the time for the prayer was due (i.e. after midday)." Volume 4, Book 53, Number 386 , Narrated Jubair bin Haiya.
. Allah's Apostle said, "Allah guarantees (the person who carries out Jihad in His Cause and nothing compelled him to go out but Jihad in His Cause and the belief in His Word) that He will either admit him into Paradise (Martyrdom) or return him with reward or booty he has earned to his residence from where he went out." Volume 9, Book 93, Number 555, Narrated Abu Huraira:

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