Part of what I like so much about both Aristotle and Brandom is that they each offer a sustained non-Subject-centered development of what I broadly think of as meta-ethical concerns, including subjectivity itself.
Since I first encountered him in the 1970s, Lacan’s broad perspective on subjectivity as decentered always seemed very sensible and right to me, and in accord with the epistemic modesty I have attributed to Plato and Aristotle. At this very broad level, there is an important consonance here.
A large emphasis on language is obviously another point Aristotle, Brandom, and Lacan all have in common, but while I have previously suggested a possibility of bidirectional translation specifically between relational structures and inferentialist modes of expression based on a common denominator of Hegelian determinate negation, the difference between metaphoric-metonymic and normative-inferential approaches to language is huge.
Lacan identified what he called the Other primarily with language analyzed in Jakobsonian terms, and with Levi-Straussian “Law”. Now I want to focus on the latter aspect. This seems to be just normativity, albeit in global synchronic relational form rather than the fine-grained interactive diachronic form developed by Brandom.
To say that the unconscious is the discourse of the Other in this sense results in an anonymous, social unconscious rather than a personal, biological one, which I find highly intriguing. It also puts the unconscious and normativity in the same “place”. At first that made me worry about explaining primary process from a Brandomian point of view, but I have decided there is no requirement to do that. I am inclining toward a view that primary process and normativity would each pick out aspects of what goes on in that hypothetical anonymous, role-based social subjective “place”, and that those aspects would be basically orthogonal.
The Other seems to be mainly considered as synchronic and global, viewed from a distance, whereas mutual recognition is a fine-grained, ongoing interactive process. The Other could be hypothetically considered as the global synchronic product of the whole mutual recognition process. Conversely, mutual recognition could be considered a detailed, internal, genetic explanation of the Other that Lacan never contemplated.
I have yet to find mutuality in Lacan. He talked a fair amount about Hegel, but through a Kojèvian lens. Kojève had stressed the struggle for one-sided recognition associated with Mastery. Lacan seems to have regarded love as primarily a narcissistic phenomenon, which I cannot agree with. (See also Imaginary, Symbolic, Real.)