For a last note on elephants, both general and specific:
The earliest evidence for Elephants as work animals is from the Harappan or Indus Valley civilization, around 2000 BCE.
The earliest evidence for elephants as war animals is much later, 550 - 540 BCE in the Magadha Kingdom of central - northern India and by the Derbices, a tribe living east of the Caspian Sea who used them against Cyrus the Great of Persia - and who got their elephants from India. A bit later, in 506 BCE the Ch'u state in southern China used elephants in battle. These were all Asian ("Indian") elephants, (Elephas maximus) which therefore have the longest history of working with humans (Note that since elephants are tamed and trained, but rarely bred in captivity except in modern zoos, they are not considered Domesticated like dogs, cats, cattle, sheep, pigs, or horses)
It isn't until about 326 BCE that there is any evidence for towers or 'howdahs' on the war elephants' backs: before that men simply sat astride the elephant's back with no protection or cover.
324 BCE Alexander the Great brings hundreds of elephants back to Babylon from India and the Diadochi ("Successors" - Seleucids, Lysimachids, etc) start using them in 'western' warfare by 321 BCE and with towers to protect the men on them by 317 BCE. Having no direct access to India, Ptolemaic Egypt starts using the North African elephant species a few years later.
Virtually everyone who could get their hands on them used elephants, including Pyrrhus of Epirus, who introduced them to Rome and the Italian states and, by 278 BCE, to Carthage, who also started catching North African elephants and training them, and had its own 'elephant corps' by 264 BCE.
The first evidence of elephant armor, in the form of large iron plates, doesn't appear until the first century BCE, in northwestern India.
By 500 CE the White Huns (Ephthalite) of Central Asia are said to have had over 2000 elephants in their armies, all obtained by trading with India.
By 1023 CE Mahmud of the Ghaznavids in Afghanistan has over 1300 elephants and among the missile weapons used by the elephant crews are 'naptha' grenades (incendiaries), In the same century the Khmer of (modern) Cambodia are using double-bow siege crossbows from elephant-back.
1388 CE saw the beginning of the end of elephants in battle when a charge by elephants was stopped by MIng Chinese gunpowder weapons firing in volleys - one of the first instances of the use of that technique with gunpowder firearms (the Chinese were already using volley firing with crossbows from multiple ranks)