A slightly off-target answer, but possibly of some use. First, indeed, there is a large population who deliberately take the viewpoint, attributed to Harish-Chandra, that everything should be done for arbitrary reductive/semi-simple groups, not just "examples". Thus, as @LSpice's comment, "there's just one reductive group, $G$". :)

But, with some substance for the theory of integral representations of automorphic $L$-functions, the obvious/natural relations of classical groups prove non-trivial theorems... that do not have "intrinsic" analogues, in any useful sense, apparently.

So! "The classical groups". Over $\mathbb C$, there are really just $3$: general or special linear, orthogonal (nevermind the parity of dimension), and symplectic. Over $\mathbb R$ we have the subdivision...

Over $\mathbb R$, we can have Sylvester's Inertia theorems for the real forms that are described by "signatures", the simplest ones being $O(p,q)$, $U(p,q)$, $Sp^*(p,q)$. Yes, modeled by reals, complex, and quaternions. In light of Weil's "Classical groups/algebras with involutions", this is not a coincidence...

So, operationally, there are two types of classical groups: general-linear, and isometry (or similitude). In general-linear, parabolics are stabilizers of flags. In isometry... groups, parabolics are stabilizers of totally isotropic flags.

Choice of Levi component is equivalent to choice of complementary isotropic flag.

One can continue, as desired, to be able to re-specify "roots", etc.

Witt's theorem(s) show that there is only one isomorphism class of choices...

The "physicality" of Inertia Theorems as a part of "classification over $\mathbb R$" is (to my mind) use of inequalities, the intermediate value theorem, and such. Yes, this is all subsumed by the less-"physical" Witt-theorem business, but some of it is (to my mind) much easier to believe.

Yes, for example, Witt's Thm assures us that the leftover, after removing $W\oplus W'$ for maximal totally isotropic $W$ and $W'$ complementing each other, is independent of choices.

The latter bit is perhaps the easiest thing to talk about from a classical-groups viewpoint, but a bit clumsy (so far as I can understand) instrinsically. Anisotropic groups in the Levi component(s) of minimal ($\mathbb R$-) parabolics.

(I must confess that the times I've tried to give courses on "Lie theory" or "algebraic groups" emphasizing "classical groups", I've met considerable resistance from a considerable fraction of the audience, who could not find corroboration for the viewpoint "in the wild".)

After quite a few years of looking at reductive groups from both ends... I do finally think that even the basic facts are fairly well determined by the smallish number of data points we have from the split and quasi-split classical groups. Perhaps more usefully, the phenomena beyond that are easy to exemplify among classical groups, but somewhat clumsy to get-under-the-umbrella of intrinsic description, it seems to me.

(A funny business...)

A slightly off-target answer, but possibly of some use. First, indeed, there is a large population who deliberately take the viewpoint, attributed to Harish-Chandra, that everything should be done for arbitrary reductive/semi-simple groups, not just "examples". Thus, as @LSpice's comment, "there's just one reductive group, $G$". :)

But, with some substance for the theory of integral representations of automorphic $L$-functions, the obvious/natural relations of classical groups prove non-trivial theorems... that do not have "intrinsic" analogues, in any useful sense, apparently.

So! "The classical groups". Over $\mathbb C$, there are really just $3$: general or special linear, orthogonal (nevermind the parity of dimension), and symplectic. Over $\mathbb R$ we have the subdivision...

Over $\mathbb R$, we can have Sylvester's Inertia theorems for the real forms that are described by "signatures", the simplest ones being $O(p,q)$, $U(p,q)$, $Sp^*(p,q)$. Yes, modeled by reals, complex, and quaternions. In light of Weil's "Classical groups/algebras with involutions", this is not a coincidence...

So, operationally, there are two types of classical groups: general-linear, and isometry (or similitude). In general-linear, parabolics are stabilizers of flags. In isometry... groups, parabolics are stabilizers of totally isotropic flags.

Choice of Levi component is equivalent to choice of complementary isotropic flag.

One can continue, as desired, to be able to re-specify "roots", etc.

Witt's theorem(s) show that there is only one isomorphism class of choices...

The "physicality" of Inertia Theorems as a part of "classification over $\mathbb R$" is (to my mind) use of inequalities, the intermediate value theorem, and such. Yes, this is all subsumed by the less-"physical" Witt-theorem business, but some of it is (to my mind) much easier to believe.

Yes, for example, Witt's Thm assures us that the leftover, after removing $W\oplus W'$ for maximal totally isotropic $W$ and $W'$ complementing each other, is independent of choices.

The latter bit is perhaps the easiest thing to talk about from a classical-groups viewpoint, but a bit clumsy (so far as I can understand) instrinsically. Anisotropic groups in the Levi component(s) of minimal ($\mathbb R$-) parabolics.

(I must confess that the times I've tried to give courses on "Lie theory" or "algebraic groups" emphasizing "classical groups", I've met considerable resistance from a considerable fraction of the audience, who could not find corroboration for the viewpoint "in the wild".)

After quite a few years of looking at reductive groups from both ends... I do finally think that even the basic facts are fairly well determined by the smallish number of data points we have from the split and quasi-split classical groups. Perhaps more usefully, the phenomena beyond that are easy to exemplify among classical groups, but somewhat clumsy to get-under-the-umbrella of intrinsic description, it seems to me.

(A funny business...)

(To be clearer: A. Weil's "Algebras with involutions and the classical groups"...)

A slightly off-target answer, but possibly of some use. First, indeed, there is a large population who deliberately take the viewpoint, attributed to Harish-Chandra, that everything should be done for arbitrary reductive/semi-simple groups, not just "examples". Thus, as @LSpice's comment, "there's just one reductive group, $G$". :)

But, with some substance for the theory of integral representations of automorphic $L$-functions, the obvious/natural relations of classical groups prove non-trivial theorems... that do not have "intrinsic" analogues, in any useful sense, apparently.

So! "The classical groups". Over $\mathbb C$, there are really just $3$: general or special linear, orthogonal (nevermind the parity of dimension), and symplectic. Over $\mathbb R$ we have the subdivision...

Over $\mathbb R$, we can have Sylvester's Inertia theorems for the real forms that are described by "signatures", the simplest ones being $O(p,q)$, $U(p,q)$, $Sp^*(p,q)$. Yes, modeled by reals, complex, and quaternions. In light of Weil's "Classical groups/algebras with involutions", this is not a coincidence...

So, operationally, there are two types of classical groups: general-linear, and isometry (or similitude). In general-linear, parabolics are stabilizers of flags. In isometry... groups, parabolics are stabilizers of totally isotropic flags.

Choice of Levi component is equivalent to choice of complementary isotropic flag.

One can continue, as desired, to be able to re-specify "roots", etc.

Witt's theorem(s) show that there is only one isomorphism class of choices...

The "physicality" of Inertia Theorems as a part of "classification over $\mathbb R$" is (to my mind) use of inequalities, the intermediate value theorem, and such. Yes, this is all subsumed by the less-"physical" Witt-theorem business, but some of it is (to my mind) much easier to believe.

Yes, for example, Witt's Thm assures us that the leftover, after removing $W\oplus W'$ for maximal totally isotropic $W$ and $W'$ complementing each other, is independent of choices.

(I must confess that the times I've tried to give courses on "Lie theory" or "algebraic groups" emphasizing "classical groups", I've met considerable resistance from a considerable fraction of the audience, who could not find corroboration for the viewpoint "in the wild".)

After quite a few years of looking at reductive groups from both ends... I do finally think that even the basic facts are fairly well determined by the smallish number of data points we have from the split and quasi-split classical groups. Perhaps more usefully, the phenomena beyond that are easy to exemplify among classical groups, but somewhat clumsy to get-under-the-umbrella of intrinsic description, it seems to me.

(A funny business...)

A slightly off-target answer, but possibly of some use. First, indeed, there is a large population who deliberately take the viewpoint, attributed to Harish-Chandra, that everything should be done for arbitrary reductive/semi-simple groups, not just "examples". Thus, as @LSpice's comment, "there's just one reductive group, $G$". :)

But, with some substance for the theory of integral representations of automorphic $L$-functions, the obvious/natural relations of classical groups prove non-trivial theorems... that do not have "intrinsic" analogues, in any useful sense, apparently.

So! "The classical groups". Over $\mathbb C$, there are really just $3$: general or special linear, orthogonal (nevermind the parity of dimension), and symplectic. Over $\mathbb R$ we have the subdivision...

Over $\mathbb R$, we can have Sylvester's Inertia theorems for the real forms that are described by "signatures", the simplest ones being $O(p,q)$, $U(p,q)$, $Sp^*(p,q)$. Yes, modeled by reals, complex, and quaternions. In light of Weil's "Classical groups/algebras with involutions", this is not a coincidence...

So, operationally, there are two types of classical groups: general-linear, and isometry (or similitude). In general-linear, parabolics are stabilizers of flags. In isometry... groups, parabolics are stabilizers of totally isotropic flags.

Choice of Levi component is equivalent to choice of complementary isotropic flag.

One can continue, as desired, to be able to re-specify "roots", etc.

Witt's theorem(s) show that there is only one isomorphism class of choices...

The "physicality" of Inertia Theorems as a part of "classification over $\mathbb R$" is (to my mind) use of inequalities, the intermediate value theorem, and such. Yes, this is all subsumed by the less-"physical" Witt-theorem business, but some of it is (to my mind) much easier to believe.

Yes, for example, Witt's Thm assures us that the leftover, after removing $W\oplus W'$ for maximal totally isotropic $W$ and $W'$ complementing each other, is independent of choices.

The latter bit is perhaps the easiest thing to talk about from a classical-groups viewpoint, but a bit clumsy (so far as I can understand) instrinsically. Anisotropic groups in the Levi component(s) of minimal ($\mathbb R$-) parabolics.

(I must confess that the times I've tried to give courses on "Lie theory" or "algebraic groups" emphasizing "classical groups", I've met considerable resistance from a considerable fraction of the audience, who could not find corroboration for the viewpoint "in the wild".)

After quite a few years of looking at reductive groups from both ends... I do finally think that even the basic facts are fairly well determined by the smallish number of data points we have from the split and quasi-split classical groups. Perhaps more usefully, the phenomena beyond that are easy to exemplify among classical groups, but somewhat clumsy to get-under-the-umbrella of intrinsic description, it seems to me.

(A funny business...)