Our talk will consider two historical case studies of the complex and contested disciplinary definition of algebra. Both the historical actors and the historians and mathematicians who look back on the history of algebra in order to define this discipline have had to make choices that are often based on tacit assumptions (e.g. assumptions that determine contextually situated answers to questions such as: what is algebra for? when is it used? does it have its own language? does it have its own objects?). Indeed, historians and mathematicians often “thematise” their narratives of mathematics' past around disciplines, and such thematisation can have distorting anachronistic effects, projecting our disciplinary definitions onto past texts. The first case study is Isaac Newton's approach to algebra. A study of the context in which Newton was engaged allows us to gain a better understanding of the tasks and values he set himself when practising algebra. These tasks and values differ in some respects from those that shape the historical narratives of 17th-century mathematics. A second case study focuses on the Abel–Ruffini theorem, which established the impossibility of solving general polynomial equations of degree five or higher. We argue that this theorem not only marked a mathematical breakthrough but also contributed to a fundamental reconfiguration of algebra's objects and boundaries as a discipline, while exploring the role that the concept of commutativity may have played in this transformation. Taken together, these case studies reveal strikingly different conceptions of algebra across time. In short, our talk is an exercise in contextualising algebra as a discipline, where the contexts are both those of the historical actors and those of present-day historians and mathematicians interpreting (and editing) past texts. Such an interplay between the categories of historical actors and those of present-day historians and mathematicians invites a dialogue between the historian and mathematician, a dialogue that helps to expose the problematic and contingent assumptions that underlie the definition(s) of algebra.