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For a real basis matrix $E$ with basis vectors as rows, the reciprocal basis has row matrix $E^{-T}$. If either basis is orthonormal, both are, and $E^{-T}=E$. Thus $E$ is an orthogonal matrix, with determinant either $1$ or $-1$. It is a rotation matrix precisely when the original ordered basis is positively oriented relative to the reference coordinates. For example, $\mathrm{diag}(1,1,-1)$ is its own orthonormal reciprocal basis matrix but is an improper orthogonal transformation.

The orthogonal group isThe special orthogonal group is its determinant-one subgroup

Every eigenvalue of an orthogonal matrix has modulus one. A real three-by-three matrix has at least one real eigenvalue, and nonreal eigenvalues occur in a complex conjugate pair. If $Q\in SO(3)$ has such a pair $e^{\pm i\theta }$, their product is one, so the remaining eigenvalue is $\det Q=1$. If all three eigenvalues are real, each is $\pm 1$ and their product is one; an odd number of three signs with product one must include $+1$. Hence every element of $SO(3)$ has an eigenvector of eigenvalue one and represents a rotation in three dimensions about its span.

It is false that every element of $O(3)$ is either a rotation or a plane reflection. For example,is an improper orthogonal transformation combining a rotation with a reflection. Its eigenvalues are $e^{\pm i\theta }$ and $-1$, whereas a plane reflection has eigenvalues $1,1,-1$ and a rotation has determinant one.

Label the vertices of a regular tetrahedron $1,2,3,4$, and include all its Euclidean isometries, both rotations and reflections. The following list describes every subgroup of cardinality at most six, allowing arbitrary relabelling of the vertices. The numbers count actual subgroups, rather than just their group isomorphism types.

Here is why this is exhaustive. Part b leaves only the trivial group, $C_2,C_3,C_4,C_2\times C_2,S_3$. The disjoint cycle decomposition gives exactly the two listed kinds of involutions, the three-cycles, and the four-cycles, accounting for all the cyclic subgroups. In a Klein four-group, the nonidentity elements commute. If it contains $(12)$, any other nonidentity element must be a commuting involution, hence $(34)$ or $(12)(34)$; either choice yields the displayed opposite-edge subgroup. If it has no transposition, its three nonidentity elements must be all three double transpositions, yielding the rotational subgroup.

Finally, an $S_3$ subgroup has a unique cyclic subgroup of cardinality three. Its two three-cycles fix the same unique vertex. Every element of the $S_3$ subgroup preserves this vertex, since conjugation preserves that unique cyclic subgroup. Thus it is the full stabilizer subgroup of that vertex. The counts in the list are consequently exhaustive, givingThe two kinds of $C_2$ and the two kinds of Klein four-groups have different geometric behavior despite having the same abstract group isomorphism type within each pair. This is the distinction recorded by the small subgroups of tetrahedral symmetry.

The full symmetry group of a regular tetrahedron is the symmetric group $S_4$ acting on its vertices. Its subgroups with at most six elements have eight types up to conjugation, with the following representative subgroups and counts:A transposition is a reflection through the plane containing the two fixed vertices and the midpoint of the exchanged edge. A double transposition is a half-turn about the axis joining the midpoints of the two exchanged edges. A three-cycle is a one-third-turn about a vertex-to-opposite-face axis. A four-cycle is a quarter-turn composed with a reflection perpendicular to the opposite-edge midpoint axis; it is an improper orthogonal transformation.

The unique Klein four-group containing only double transpositions consists of the three opposite-edge half-turns and the identity element. The other three Klein four-groups consist of two perpendicular mirror reflections, their half-turn product, and the identity element. Each $S_3$ is a stabilizer subgroup of a vertex, including the three rotations and three mirror reflections preserving that vertex.

For completeness, a Klein four-group containing $(12)$ must contain the commuting involution $(34)$ or $(12)(34)$, yielding the displayed subgroup; one containing no transpositions must contain all three double transpositions. An $S_3$ subgroup has a unique cyclic subgroup of cardinality three, whose unique fixed vertex must therefore be preserved by the entire $S_3$. These observations and the disjoint cycle decomposition exhaust the types allowed by groups of order at most six. The counts sum to $25$.