Such a definition (but not the definition, I suppose) can be found in Clifford Algebras and the Classical Groups by Ian Porteous (see Chapt. 13).

It is based on the classification of real algebra anti-involutions of $A(n)$ where $A$ is equal to $K$ or $^2K$, and $K = \mathbb R$, $\mathbb C$ or $\mathbb H$. By the theorem below there are ten cases. In each case there is a corresponding family of groups of correlated automorphisms analogous to the orthogonal groups.

Theorem. Let $\xi$ be an irreducible correlation on a right $A$-linear space of finite dimension $> 1$, and therefore equivalent to a symmetric or skew correlation. Then $\xi$ is equivalent to one of the following ten types, these being mutually exclusive.

1. A symmetric $\mathbb R$-correlation; 2. A symmetric, or equivalently a skew, $^2\mathbb R^\sigma$-correlation; 3. A skew $\mathbb R$-correlation; 4. A skew $\mathbb C$-correlation; 5. A skew $\tilde{\mathbb H}$- or equivalently a symmetric $\overline{\mathbb H}$-correlation; 6. A skew, or equivalently a symmetric,$^2\overline{\mathbb H}^\sigma$ -correlation; 7. A symmetric $\tilde{\mathbb H}$-, or equivalently a skew, $\overline{\mathbb H}$-correlation; 8. A symmetric $\mathbb C$-correlation; 9. A symmetric, or equivalently a skew, $\overline{\mathbb C}$-correlation; 10. A symmetric, or equivalently a skew, $^2\overline{\mathbb C}^\sigma$-correlation.

The ten families of classical groups are as follows, where $p+q=n$:

1. $O(p, q; \mathbb R)$ or $O(p, q)$, with $O(n) = 0(0, n)$; 2. $GL(n;\mathbb R)$; 3. $Sp(2n;\mathbb R)$;
   4. $Sp(2n;\mathbb C)$ 5. $Sp(p,q;\mathbb H)$ or $Sp(p,q)$, with $Sp(n)= Sp(0,n)$; 6. $GL(n;\mathbb H)$ 7. $O(n;\mathbb H)$ 8. $O(n;\mathbb C)$ 9. $U(p,q)$, with $U(n)=U(0,n);$ 10. $GL(n;\mathbb C)$.

Such a definition (but not the definition, I suppose) can be found in Clifford Algebras and the Classical Groups by Ian Porteous (see Chapt. 13).

It is based on the classification of real algebra anti-involutions of $A(n)$ where $A$ is equal to $K$ or $^2K$, and $K = \mathbb R$, $\mathbb C$ or $\mathbb H$. By the theorem below there are ten cases. In each case there is a corresponding family of groups of correlated automorphisms analogous to the orthogonal groups.

Theorem. Let $\xi$ be an irreducible correlation on a right $A$-linear space of finite dimension $> 1$, and therefore equivalent to a symmetric or skew correlation. Then $\xi$ is equivalent to one of the following ten types, these being mutually exclusive.

1. A symmetric $\mathbb R$-correlation; 2. A symmetric, or equivalently a skew, $^2\mathbb R^\sigma$-correlation; 3. A skew $\mathbb R$-correlation; 4. A skew $\mathbb C$-correlation; 5. A skew $\tilde{\mathbb H}$- or equivalently a symmetric $\overline{\mathbb H}$-correlation; 6. A skew, or equivalently a symmetric,$^2\overline{\mathbb H}^\sigma$ -correlation; 7. A symmetric $\tilde{\mathbb H}$-, or equivalently a skew, $\overline{\mathbb H}$-correlation; 8. A symmetric $\mathbb C$-correlation; 9. A symmetric, or equivalently a skew, $\overline{\mathbb C}$-correlation; 10. A symmetric, or equivalently a skew, $^2\overline{\mathbb C}^\sigma$-correlation.

The ten families of classical groups are as follows, where $p+q=n$:

1. $O(p, q; \mathbb R)$ or $O(p, q)$, with $O(n) = 0(0, n)$; 2. $GL(n;\mathbb R)$; 3. $Sp(2n;\mathbb R)$;
   4. $Sp(2n;\mathbb C)$; 5. $Sp(p,q;\mathbb H)$ or $Sp(p,q)$, with $Sp(n)= Sp(0,n)$; 6. $GL(n;\mathbb H)$; 7. $O(n;\mathbb H)$; 8. $O(n;\mathbb C)$; 9. $U(p,q)$, with $U(n)=U(0,n)$; 10. $GL(n;\mathbb C)$.

Such a definition (but not the definition, I suppose) can be found in Clifford Algebras and the Classical Groups by Ian Porteous (see Chapt. 13).

It is based on the classification of the anti-involutions of $A(n)$ where $A$ is equal to $K$ or $^2K$, and $K = \mathbb R$, $\mathbb C$ or $\mathbb H$. By the theorem below there are ten cases. In each case there is a corresponding family of groups of correlated automorphisms analogous to the orthogonal groups.

Theorem. Let $\xi$ be an irreducible correlation on a right $A$-linear space of finite dimension $> 1$, and therefore equivalent to a symmetric or skew correlation. Then $\xi$ is equivalent to one of the following ten types, these being mutually exclusive.

1. A symmetric $\mathbb R$-correlation; 2. A symmetric, or equivalently a skew, $^2\mathbb R^\sigma$-correlation; 3. A skew $\mathbb R$-correlation; 4. A skew $\mathbb C$-correlation; 5. A skew $\tilde{\mathbb H}$- or equivalently a symmetric $\overline{\mathbb H}$-correlation; 6. A skew, or equivalently a symmetric,$^2\overline{\mathbb H}^\sigma$ -correlation; 7. A symmetric $\tilde{\mathbb H}$-, or equivalently a skew, $\overline{\mathbb H}$-correlation; 8. A symmetric $\mathbb C$-correlation; 9. A symmetric, or equivalently a skew, $\overline{\mathbb C}$-correlation; 10. A symmetric, or equivalently a skew, $^2\overline{\mathbb C}^\sigma$-correlation.

The ten families of classical groups are as follows, where $p+q=n$:

1. $O(p, q; \mathbb R)$ or $O(p, q)$, with $O(n) = 0(0, n)$; 2. $GL(n;\mathbb R)$; 3. $Sp(2n;\mathbb R)$;
   4. $Sp(2n;\mathbb C)$ 5. $Sp(p,q;\mathbb H)$ or $Sp(p,q)$, with $Sp(n)= Sp(0,n)$; 6. $GL(n;\mathbb H)$ 7. $O(n;\mathbb H)$ 8. $O(n;\mathbb C)$ 9. $U(p,q)$, with $U(n)=U(0,n);$ 10. $GL(n;\mathbb C)$.

Such a definition (but not the definition, I suppose) can be found in Clifford Algebras and the Classical Groups by Ian Porteous (see Chapt. 13).

It is based on the classification of real algebra anti-involutions of $A(n)$ where $A$ is equal to $K$ or $^2K$, and $K = \mathbb R$, $\mathbb C$ or $\mathbb H$. By the theorem below there are ten cases. In each case there is a corresponding family of groups of correlated automorphisms analogous to the orthogonal groups.

Theorem. Let $\xi$ be an irreducible correlation on a right $A$-linear space of finite dimension $> 1$, and therefore equivalent to a symmetric or skew correlation. Then $\xi$ is equivalent to one of the following ten types, these being mutually exclusive.

1. A symmetric $\mathbb R$-correlation; 2. A symmetric, or equivalently a skew, $^2\mathbb R^\sigma$-correlation; 3. A skew $\mathbb R$-correlation; 4. A skew $\mathbb C$-correlation; 5. A skew $\tilde{\mathbb H}$- or equivalently a symmetric $\overline{\mathbb H}$-correlation; 6. A skew, or equivalently a symmetric,$^2\overline{\mathbb H}^\sigma$ -correlation; 7. A symmetric $\tilde{\mathbb H}$-, or equivalently a skew, $\overline{\mathbb H}$-correlation; 8. A symmetric $\mathbb C$-correlation; 9. A symmetric, or equivalently a skew, $\overline{\mathbb C}$-correlation; 10. A symmetric, or equivalently a skew, $^2\overline{\mathbb C}^\sigma$-correlation.

The ten families of classical groups are as follows, where $p+q=n$:

1. $O(p, q; \mathbb R)$ or $O(p, q)$, with $O(n) = 0(0, n)$; 2. $GL(n;\mathbb R)$; 3. $Sp(2n;\mathbb R)$;
   4. $Sp(2n;\mathbb C)$ 5. $Sp(p,q;\mathbb H)$ or $Sp(p,q)$, with $Sp(n)= Sp(0,n)$; 6. $GL(n;\mathbb H)$ 7. $O(n;\mathbb H)$ 8. $O(n;\mathbb C)$ 9. $U(p,q)$, with $U(n)=U(0,n);$ 10. $GL(n;\mathbb C)$.