= Solution
Let $K_i$ generate <Lorentz boosts> with the same exponential convention as the rotation <generators of rotations> $J_i$. The <Lie algebra of the Lorentz group> is
$$
[J_i,J_j]=i\epsilon_{ijk}J_k,\qquad
[J_i,K_j]=i\epsilon_{ijk}K_k,\qquad
\boxed{[K_i,K_j]=-i\epsilon_{ijk}J_k.}
$$
Consequently non-collinear <Lorentz boosts> do not close among themselves: the product contains a <rotation>, the <Wigner rotation>. For an explicit counterexample, in coordinates $(ct,x,y)$ write $c_a=\cosh a$, $s_a=\sinh a$ and similarly for $b$. Then
$$
B_y(b)B_x(a)=\begin{pmatrix}
c_bc_a&-c_bs_a&-s_b\\
-s_a&c_a&0\\
-s_bc_a&s_bs_a&c_b
\end{pmatrix}.
$$
A pure rotationless <Lorentz boost> has a symmetric <matrix>, whereas this product has spatial entries $(xy)=0$ and $(yx)=s_as_b\ne0$ for $a,b\ne0$. It is therefore a concrete product of two pure <Lorentz boosts> that is not a pure <Lorentz boost>. Collinear <Lorentz boosts> are the special case that do form a one-parameter <group>, with additive <rapidities>.
Allowing all spatial directions and closure generates both the $K_i$ and, through their <commutators>, all $J_i$. The smallest connected <group> containing all these <Lorentz boosts> is the <proper orthochronous Lorentz group>
$$
\boxed{SO^+(1,3)=\{\Lambda:\Lambda^Tg\Lambda=g,\ \det\Lambda=+1,\ \Lambda^0{}_0\geq1\}.}
$$
It contains arbitrary compositions of spatial <rotations> and <Lorentz boosts> continuously connected to the identity. It has six continuous parameters. Here “smallest” refers to the subgroup generated by all continuous pure <Lorentz boosts>, not to an arbitrary collection already including discrete reversals.
The full <Lorentz group> $O(1,3)$ has four connected components, distinguished by the sign of the <determinant> and the direction of time. Representatives are the identity, <parity> $P=\operatorname{diag}(1,-1,-1,-1)$, <time reversal> $\mathsf T=\operatorname{diag}(-1,1,1,1)$, and $P\mathsf T=-I$. Thus
$$
O(1,3)=SO^+(1,3)\ \cup\ P SO^+(1,3)\ \cup\ \mathsf T SO^+(1,3)\ \cup\ P\mathsf T SO^+(1,3).
$$
Multiplication by these discrete transformations relates the components algebraically; no continuous path within the <Lorentz group> joins different components. Translations are not required for closure here; adding them gives the larger <Poincaré group>.
Back to article page
