$\mathfrak{sl}\_2(\mathbb{C})$ is usually given a basis $H, X, Y$ satisfying $[H, X] = 2X, [H, Y] = -2Y, [X, Y] = H$. What is the origin of the use of the letter $H$? (It certainly doesn't stand for "Cartan.") My guess, based on similarities between these commutator relations and ones I have seen mentioned when people talk about physics, is that $H$ stands for "Hamiltonian." Is this true? Even if it's not, is there a connection? 



What's the origin of the naming convention for the standard basis of $\mathfrak{sl}_2(\mathbb{C}) $?
The letters $\mathrm X$ and $\mathrm Y$ are already used by **Cayley** in what Dieudonné (in [MR](//ams.org/mathscinet-getitem?mr=88c:01020)) calls the first description of all finite-dimensional irreducible $\mathfrak{sl}\_2$-modules: *A Second Memoir upon Quantics* ([1856](//doi.org/10.1098/rstl.1856.0008), §§29–31). He apparently has no name for $\mathrm{XY-YX}$. Same in e.g. Faà di Bruno ([1876](//zbmath.org/?q=an:08.0056.02), [§§113–114](//archive.org/details/thoriedesformes00brungoog/page/n218)).


$\mathrm H$ seems to stand for *Hauptmatrix*, as introduced by **Weyl** ([1922](//zbmath.org/?q=an:48.0637.04), p. 125; see also [1931](//zbmath.org/?q=an:0001.17501), pp. [114](//books.google.com/books?id=jBDOAAAAMAAJ&&q=%22welche+die+beiden+Variablen+einzeln+transformieren%22), [122](//books.google.com/books?id=jBDOAAAAMAAJ&q=Haupttransformationen)) to describe Cartan subalgebras, roots (*Multiplikatoren*) and root spaces (*Länder*): 



> 
> Kommt in $\mathfrak g$ eine „Hauptmatrix“ $H$ vor, in der alle Elemente außerhalb der Hauptdiagonale verschwinden, während in der Hauptdiagonale die Zahlen $\alpha\_1,\alpha\_2,\dots,\alpha\_n$ stehen, so bilde man die Differenzen $\alpha\_i - \alpha\_k$ und teile mit Bezug auf $H$ das Schema einer beliebigen Matrix in „Länder“ ein, indem man jedem Feld $(ik)$ des Schemas ($i$ der Zeilen-, $k$ der Kolonnenindex) die Zahl $\alpha\_i - \alpha\_k$ als „Multiplikator“ zuordnet (...)
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> 


$\mathfrak{sl}\_2$-triples with your bracket relations appear in Killing ([1888](//zbmath.org/?q=an:20.0368.03), p. 281), denoted $(X\_{r-1}, X\_r, X\_{r-2})$; Cartan ([1894](//zbmath.org/?q=an:25.0638.02), [p. 116](//archive.org/details/surlastructured00bourgoog/page/n125)), denoted $\mathrm{(Y, X,X')}$; Weyl ([1925](//zbmath.org/?q=an:51.0319.01), p. 276), denoted $(h\_\alpha,e\_\alpha,e\_{-\alpha})$, with the $h\_\alpha$ again called “Diagonal- oder Hauptmatrizen”; Dynkin ([1952](http://mi.mathnet.ru/eng/msb/v72/i2/p349), §8.1), denoted $(f,e\_+,e\_-)$; **Chevalley** ([1955](//zbmath.org/?q=an:0066.01503), p. 28; [1955](//ams.org/mathscinet-getitem?mr=68552), [p. 96](//books.google.com/books?id=YjfvAAAAMAAJ&q=%22%5BH,+X%5D+%3D+aX%22)), denoted $(D,N,N')$, $(H\_r,X\_r,X\_{-r})$, and finally the desired $\mathrm{(H,X,Y)}$. 


(Standardization was slow: Lie ([1876](//zbmath.org/?q=an:08.0212.01), [p. 53](//archive.org/details/archivformathema11876oslo/page/53); [1890](//zbmath.org/?q=an:23.0364.01), [p. 353](//archive.org/details/theotransformation02liesrich/page/n366)) used $(X\_1,X\_2,X\_3)=(\mathrm{-Y,\smash{\frac12}H,X})$, and similarly rescaled bases and bracket relations still appear in Pauli ([1927](//doi.org/10.1007/BF01397326), p. 614), Born-Jordan ([1930](//zbmath.org/?q=an:56.1291.01), p. 135), Casimir-van der Waerden ([1931](https://zbmath.org/?q=an:57.1579.02), [p. 46](//www.lorentz.leidenuniv.nl/IL-publications/dissertations/ehrenfest.html); [1935](//zbmath.org/?q=an:61.0475.02), p. 4), Bauer ([1933](//zbmath.org/?q=an:59.1521.02), p. 126), Harish-Chandra ([1950](//ams.org/mathscinet-getitem?mr=33811), p. 301; [1952](//ams.org/mathscinet-getitem?mr=47055), p. 337), Jacobson ([1951](//ams.org/mathscinet-getitem?mr=49882), p. 107; [1958](//ams.org/mathscinet-getitem?mr=20:3901), p. 825), Séminaire “Sophus Lie” ([1955](//ams.org/mathscinet-getitem?mr=73107), [p. 10-01](http://www.numdam.org/item/SSL_1954-1955__1__A14_0)), Kostant ([1959](//ams.org/mathscinet-getitem?mr=22:5693), p. 977), etc. Settling on the “Chevalley” basis $\mathrm{(H,X,Y)}$ over Lie’s seems ultimately motivated by the smaller $\mathbf Z$-form $\mathfrak g\_\mathrm{sc}\subset\mathfrak g\_\mathrm{ad}$ it spans, cf. Borel ([1970](//ams.org/mathscinet-getitem?mr=258838), §2.7).) 


