Film-maker Lilly Wachowski has responded to Elon Musk and Ivanka Trump after the pair bantered on social media using a phrase from the Wachowskis’ hit 1999 film, The Matrix.

After Musk tweeted “Take the red pill” – referring to a key scene in The Matrix, which subsequently gave rise to the phrase “redpilling” as an alt-right, misogynist meme, along with a red rose emoji normally associated with the Democratic Socialists of America, the political organisation to which Alexandria Ocasio-Cortez and Rashida Tlaib belong – Trump replied: “Taken!”

Wachowski then pithily responded, “Fuck both of you”, before following up with a tweet supporting the Brave Space Alliance, which describes itself as “the first Black and trans led LGBTQIA center located on the South Side of Chicago”.

Musk, the South African-born entrepreneur who is CEO of SpaceX and Tesla, posted his original tweet with no other comment, but it is thought to refer to his clash with authorities in California over his plans to reopen the Tesla factory in Fremont, in which he was supported by Donald Trump.

In The Matrix, the selection of blue or red pill offered by Laurence Fishburne’s revolutionary leader Morpheus to reluctant-hero hacker Neo (played by Keanu Reeves) is framed as a choice between the blue pill’s wilful ignorance – “You wake up in your bed and believe whatever you want to believe” or the red pill’s introduction to “the truth”: “I show you how deep the rabbit hole goes.”

Wachowski came out as trans in 2016, following fellow Matrix director and sibling Lana, who came out in 2012.

• This article was amended on 20 May 2020 because an earlier version referred to the Democratic Socialists of America as a “party” whereas it is a political organisation.