First of all, I'd inquire what role the characteristic of the field plays here. It's true that the finite twisted groups rely on characteristics 2, 3 especiallu, but infinite twisted groups include the special unitary groups (etc.). which live over many fields.

While steinberg's lecture notes are often quite useful, a more leisurely treatment (often with more detail) is found in Roger Carter's 1972 book Simple Groups of Lie Type here .

In any case, the groups $\hat{H}$ and the like are tori in the algebraic group sense and thus easy to describe abstractly. But in this context they take a little more work to pin down. The best strategy is probably to follow Carter's treatment for the specific case of unitary groups over an infinite field.

ADDED: First I'll mention some older papers, now freely available in PDF format for online use. These can give some idea of the arguments used as well as how the notation has shifted. First is Chevalley's important 1955 paper in the Tohoku Math. J. Next is Steinberg's 1959 paper Variations on a theme of Chevalley in the Pacific J. Math. (with independent contributons along the same line in a paper by J. Tits and a thesis by David Hertzig at Chicago directed I think by Andre Weil).

For Steinberg's 1960 treatment of automorphisms of Chevalley groups(etc.) over finite fields, see Canadian J. Math., while my follow-up paper on working over arbitrary infinite fields is in the same journal, here .

Aside from differences in notation, the algebraic torus $\widehat{H}$ for a Chevalley group should lead to the torus for a twisted group of type $A_n, n \geq 2$ or $D_n$ or $E_6$ given by the fixed points of the graph automorphism involved combined with a field automorphism of the same order.

First of all, I'd inquire what role the characteristic of the field plays here. It's true that the finite twisted groups rely on characteristics 2, 3 especiallu, but infinite twisted groups include the special unitary groups (etc.). which live over many fields.

While Steinberg's lecture notes are often quite useful, a more leisurely treatment (often with more detail) is found in Roger Carter's 1972 book Simple Groups of Lie Type here .

In any case, the groups $\hat{H}$ and the like are tori in the algebraic group sense and thus easy to describe abstractly. But in this context they take a little more work to pin down. The best strategy is probably to follow Carter's treatment for the specific case of unitary groups over an infinite field.

ADDED: First I'll mention some older papers, now freely available in PDF format for online use. These can give some idea of the arguments used as well as how the notation has shifted. First is Chevalley's important 1955 paper in the Tohoku Math. J. Next is Steinberg's 1959 paper Variations on a theme of Chevalley in the Pacific J. Math. (with independent contributons along the same line in a paper by J. Tits and a thesis by David Hertzig at Chicago directed I think by Andre Weil).

For Steinberg's 1960 treatment of automorphisms of Chevalley groups(etc.) over finite fields, see Canadian J. Math., while my follow-up paper on working over arbitrary infinite fields is in the same journal, here .

Aside from differences in notation, the algebraic torus $\widehat{H}$ for a Chevalley group should lead to the torus for a twisted group of type $A_n, n \geq 2$ or $D_n$ or $E_6$ given by the fixed points of the graph automorphism involved combined with a field automorphism of the same order.

First of all, I'd inquire what role the characteristic of the field plays here. It's true that the finite twisted groups rely on characteristics 2, 3 especiallu, but infinite twisted groups include the special unitary groups (etc.). which live over many fields.

While steinberg's lecture notes are often quite useful, a more leisurely treatment (often with more detail) is found in Roger Carter's 1972 book Simple Groups of Lie Type here .

In any case, the groups $\hat{H}$ and the like are tori in the algebraic group sense and thus easy to describe abstractly. But in this context they take a little more work to pin down. The best strategy is probably to follow Carter's treatment for the specific case of unitary groups over an infinite field.

First of all, I'd inquire what role the characteristic of the field plays here. It's true that the finite twisted groups rely on characteristics 2, 3 especiallu, but infinite twisted groups include the special unitary groups (etc.). which live over many fields.

While steinberg's lecture notes are often quite useful, a more leisurely treatment (often with more detail) is found in Roger Carter's 1972 book Simple Groups of Lie Type here .

In any case, the groups $\hat{H}$ and the like are tori in the algebraic group sense and thus easy to describe abstractly. But in this context they take a little more work to pin down. The best strategy is probably to follow Carter's treatment for the specific case of unitary groups over an infinite field.

ADDED: First I'll mention some older papers, now freely available in PDF format for online use. These can give some idea of the arguments used as well as how the notation has shifted. First is Chevalley's important 1955 paper in the Tohoku Math. J. Next is Steinberg's 1959 paper Variations on a theme of Chevalley in the Pacific J. Math. (with independent contributons along the same line in a paper by J. Tits and a thesis by David Hertzig at Chicago directed I think by Andre Weil).

For Steinberg's 1960 treatment of automorphisms of Chevalley groups(etc.) over finite fields, see Canadian J. Math., while my follow-up paper on working over arbitrary infinite fields is in the same journal, here .

Aside from differences in notation, the algebraic torus $\widehat{H}$ for a Chevalley group should lead to the torus for a twisted group of type $A_n, n \geq 2$ or $D_n$ or $E_6$ given by the fixed points of the graph automorphism involved combined with a field automorphism of the same order.