A simple Lie algebra admits a non-degenerate inner product, called the Killing form. For complex matrix Lie algebras, the Killing form can be taken to be \begin{equation} B(X,Y) = \tr(XY) \end{equation} A vector space with a non-degenerate inner product admits an orthonormal basis $\{X_m\}$ satisfying \begin{equation} B(X_m,X_n) = \pm \delta_{mn} \end{equation} where $\delta_{mn}$ denotes the Kronecker delta, which is $1$ if $m=n$ and $0$ otherwise.

Consider the Lie algebra $\so(3,1)$, corresponding to the Lorentz group $\SO(3,1)$ considered in § 6.6 and § 9.3. Since Lie algebras represent infinitesimal transformations, the question of double covers never arises. Thus, at the Lie algebra level, \begin{align} \su(2) &\cong \so(3) \\ \sl(2,\CC) &\cong \so(3,1) \end{align} Recalling that the Pauli matrices $\sigma_m$ each square to the identity matrix, and that the generators of $\su(2)$ are $\tau_m=-i\sigma_m$, we see that the Killing form is negative definite on $\su(2)$. What about the boosts? But the infinitesimal boosts in $\sl(2,\CC)$ are just the Pauli matrices themselves! We are therefore led to identify elements with negative squared Killing norm as rotations, and elements with positive squared Killing norm as boosts.

The classification of simple Lie algebras outlined in § 10.3 involved complex Lie algebras. But we have just argued that the complexification of $\su(2)$ is precisely the complex Lie algebra $\sl(2,\CC)$! Put differently, a complex Lie algebra always contains an equal number of boosts and rotations, since you can multiply by $i$ to get from one to the other.

Physical symmetry groups, however, correspond to real Lie algebras. Such Lie algebras can be represented using complex matrices; what makes them real is that the commutators between elements must all have real coefficients. These coefficients are called structure constants. As a complex Lie algebra, $\sl(2,\CC)$ has just three independent elements — and multiplication by $i$ is allowed. As a real Lie algebra, $\sl(2,\CC)$ has six independent elements — since multiplication by $i$ is not allowed.

From now on, we work exclusively with real Lie algebras. As with $\su(2)\subset\sl(2,\CC)$, every real Lie algebra can be regarded as a subalgebra of a complex Lie algebra. But how many real subalgebras does a complex Lie algebra have?

In the case of $\sl(2,\CC)$, it is easy to see that we must choose either $\sigma_x$ or $\tau_x$. If we have both, then either there are no other elements, in which case the algebra is abelian (and not simple), or it includes all of $\sl(2,\CC)$. Similar considerations apply to the $y$ and $z$ basis elements. It therefore appears that we have $2^3=8$ possible 3-dimensional real subalgebras. However, remember that the structure constants must be real; the algebra must close, with real coefficients. Due to the cyclic symmetry among $x$, $y$, and $z$, there are only two inequivalent 3-dimensional real subalgebras of $\sl(2,\CC)$, generated either by $\{\tau_x,\tau_y,\tau_z\}$ or by $\{\sigma_x,\sigma_y,\tau_z\}$. We say that there are two real forms of $\aa_1=\su(2)$, namely $\su(2)$ (all rotations), and $\su(1,1)=\so(2,1)$ (two boosts and one rotation). Since rotations have compact orbits (and boosts do not), the Lie groups corresponding to real forms containing only rotations are themselves compact. We use the same language for Lie groups: The two real forms of $\SO(3)$ are $\SO(3)$ itself, which is compact, and $\SO(2,1)$.

This idea carries over to real forms of other Lie algebras. Different real forms of a given Lie algebra must all have the same complexification, that is, they are different real subalgebras of a given complex Lie algebra. In most cases, each such real form is fully determined by giving its Killing–Cartan classification and the number of boosts it contains. Determining the allowed number of boosts requires checking under what circumstances the structure constants are real. We already know some examples: Each $\so(p,q)$ is a real form of $\so(p+q)$ (namely the one with $pq$ boosts), and each $\su(p,q)$ is a real form of $\su(p+q)$ (namely the one with $2pq$ boosts). But these are not the only real forms.