By 1957, Alexander was already radicalised and a member of the Cape Peninsula Students' Union, an affiliate of The Non-European Unity Movement of South Africa. He joined the African Peoples Democratic Union of Southern Africa which was established in 1960. However he was ejected from APDUSA in 1961 and with Dulcie September, Ottilie Abrahams, and Andreas Shipinga, among others, formed a study group of nine members in July 1962, known as the Yu Chi Chan Club ; Yu Chi Chan is the Chinese name for guerrilla warfare, which Mao Zedong used. The YCCC disbanded in late 1962 and was replaced by the National Liberation Front, which Alexander co-founded. In July 1963, he, along with most members of the NLF, was arrested. In 1964, he was convicted of conspiracy to commit sabotage. From 1964–1974 he was imprisoned on Robben Island.
Post-Apartheid
After being released Alexander did pioneering work in the field of language policy and planning in South Africa from the early 1980s via organisations such as The Project for the Study of Alternative Education in South Africa, as well as the LANGTAG process. He was influential in respect to language policy development with various government departments, including Education. His most recent work was focused on the tension between multilingualism and the hegemony of English in the public sphere. He founded and was director of PRAESA from 1992 until the end of 2011 and a member of the Interim Governing Board of the African Academy of Languages. In 1981, he was appointed Director of the South African Committee for Higher Education. At the time of his death, he had retired from being director of PRAESA at the University of Cape Town. In 1994, his Trotskyist Workers Organisation for Socialist Action contested the elections. Alexander received the Linguapax Prize for 2008. The prize is awarded annually in recognition of contributions to linguistic diversity and multilingual education. The citation noted that he had devoted more than twenty years of his professional life to defend and preserve multilingualism in the post-apartheid South Africa and had become one of the major advocates of linguistic diversity.
Death
Alexander died from cancer following a short period of ill-health on 27 August 2012, aged 75. His personal archive was donated to the University of Cape Town's Special Collections library. In 2014, the were included in the library's Manuscripts and Archives collections. The university has also named a building on its upper campus after him.