Indonesian officials took offence at teaching material found at a Perth miltary base that questioned Indonesia's sovereignty in West Papua.
The country announced it was severing some military ties, but now says the suspension will only apply to language classes.
The suspension began after Indonesia's military chief took offence at material presented at a training course at a military base in Perth.
Indonesia's Special Forces group, Kopassus trains with soldiers from Australia's Special Air Services Regiment at the base.
General Gatot Nurmantyo says the piece ridiculed Pancasila-Indonesia's founding principles.
"The curriculum and the lessons, as well as the result of the curriculum and the assignment paper of the students were insulting. It hurt so much, I don't need to explain it here. It was about an officer in the past. It was about East Timor, it was about Papua needing independence. It was about another manipulation. "
Indonesia's President Joko Widodo intervened, describing ties with Australia as being in good condition.
He's asked his defence minister and military chief to investigate.
Prime Minister Malcolm Turnbull says he looks forward to the matter being resolved as soon as possible.
Australia's Deputy Prime Minister went a step further.
"It's a very warm working relationship. I believe this issue will quickly be put aside because it's in both our interests to work together. The terrorists that come into Indonesia are the same ones that are going to come into Australia. The training we give to Indonesia is just as pertinent as the training we give to Australia."
A specialist on Australia's bilateral relationship with Indonesia, Professor Tim Lindsey, says it's often volatile, with particular sensitivity over West Papua.
The Indonesian province has long wanted to obtain independence from Jakarta.
Australia recognises Indonesian sovereignty over West Papua.
But Professor Lindsey says there's mistrust towards Australia and a misconception among many about Australia's motives.
"Many Indonesians believe, quite incorrectly, that Australian has colonial style ambitions over West Papua. They see it as a significant Christian area and one that would be for that reason attractive to Australia. This is a widespread misconception in Indonesia so activity by supporters of West Papuan independence in Australia is always a flashpoint in relationships because of that absence of trust."
Professor Lindsey says Australia needs to realise that comments which might not have been deemed offensive in the past could be regarded as just that under President Widodo, who campaigned for presidency by appealing to nationalist sentiment.
"Indonesia under President Jokowi is a country that is very sensitive to perceived slights to its status, to national sovereignty sand has now got a record under Jokowi of responding assertively and with great sensitivity to anything that it might see as insulting or offensive or threatening to its authority and sovereignty. This, in many ways relates back to rhetoric of national sovereignty which hugely escalated during the election campaign when Jokowi and Prabowo were competing for election."
Australian-based West Papuan activist, Ronny Kareni, hopes one day Australia will take an different line on West Papua and speak out against what he says is Jakarta's role in human rights abuses in the province.
"It will eventually happen and we believe that the more of the people aware of that situation in West Papua and the more people here in Australia speaking up about it, that will come out, that eventually the Australian government will not stay silent but will have to speak up or take actions in combating the human rights issue that are happening in West Papua."
