M. J. Mehl, et al. (doi=10.1016/j.commatsci.2017.01.017)

The trigonal crystal system is defined by a three-fold rotation axis,
and can be generated from the cubic crystal system
by stretching the cube along its
diagonal. The symmetry requires the primitive vectors to have the
form $a = b$, $\alpha = \beta = \pi/2$, $\gamma =
120^\circ$. (We could take $\gamma = 60^\circ$, but in that
case the three-fold rotation axis is not obvious from the
primitive vectors.) The trigonal system is a limiting case of the
simple monoclinic Bravais lattice,
with $\beta = 120^\circ$. It can also be obtained from the
base-centered orthorhombic Bravais lattice
with $b = \sqrt3 a$. The conventional unit cell is described by the vectors
\begin{eqnarray}
\label{equ:lat10c}
\mathbf{A}_1 & = & \frac{a}{2} \, \mathbf{\hat{x}} - \frac{\sqrt{3}}{2} \, a \,
\mathbf{\hat{y}} \nonumber \\
\mathbf{A}_2 & = & \frac{a}{2} \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2} \, a \,
\mathbf{\hat{y}} \nonumber \\
\mathbf{A}_3 & = & c \, \mathbf{\hat{z}}.
\end{eqnarray}
There are two Bravais lattices in the trigonal system.

Somewhat confusingly, what might be called the simple trigonal
Bravais lattice is known as the hexagonal lattice. It shares
the same primitive vectors, but not point operations, as the
hexagonal crystal system.
The primitive vectors are identical to those of the conventional cell,
\begin{eqnarray}
\label{equ:lat10}
\mathbf{a}_1 & = & \frac{a}{2} \, \mathbf{\hat{x}} - \frac{\sqrt{3}}{2} \, a \,
\mathbf{\hat{y}} \nonumber \\
\mathbf{a}_2 & = & \frac{a}{2} \, \mathbf{\hat{x}} + \frac{\sqrt{3}}{2} \, a \,
\mathbf{\hat{y}} \nonumber \\
\mathbf{a}_3 & = & c \, \mathbf{\hat{z}}.
\end{eqnarray}
The volume of the primitive cell is
\begin{equation}
V = \left(\frac{\sqrt3}{2}\right) \, a^2 \, c.
\label{equ:vol10}
\end{equation}

The space groups associated with the (trigonal) hexagonal lattice
are
\begin{array}{lll}
143. ~ \mbox{P3} & 144. ~ \mbox{P3$_{1}$} & 145. ~ \mbox{P3$_{2}$} \\
147. ~ \mbox{P$\overline{3}$} & 149. ~ \mbox{P312} & 150. ~ \mbox{P321} \\
151. ~ \mbox{P3$_{1}$12} & 152. ~ \mbox{P3$_{1}$21} & 153. ~ \mbox{P3$_{2}$12} \\
154. ~ \mbox{P3$_{2}$21} & 156. ~ \mbox{P3m1} & 157. ~ \mbox{P31m} \\
158. ~ \mbox{P3c1} & 159. ~ \mbox{P31c} & 162. ~ \mbox{P$\overline{3}$1m} \\
163. ~ \mbox{P$\overline{3}$1c} & 164. ~ \mbox{P$\overline{3}$m1} & 165. ~ \mbox{P$\overline{3}$c1} \\
\end{array}

The rhombohedral Bravais lattice has the periodicity of the
conventional trigonal cell, with the addition of
two translation vectors, $2/3 \mathbf{A}_1 + 1/3 \mathbf{A}_2 + 1/3
\mathbf{A}_3$ and $1/3 \mathbf{A}_1 + 2/3 \mathbf{A}_2 + 2/3 \mathbf{A}_3$.
The primitive vectors can be taken in the form
\begin{eqnarray}
\label{equ:lat11}
\mathbf{a}_1 & = & ~ \frac{a}{2} \, \mathbf{\hat{x}} - \frac{a}{\left(2 \sqrt{3}\right)} \, \mathbf{\hat{y}} + \frac{c}{3}
\, \mathbf{\hat{z}} \nonumber \\
\mathbf{a}_2 & = & \frac{a}{\sqrt{3}} \, \mathbf{\hat{y}} + \frac{c}{3} \, \mathbf{\hat{z}} \nonumber \\
\mathbf{a}_3 & = & - \frac{a}{2} \, \mathbf{\hat{x}} - \frac{a}{\left(2 \sqrt{3}\right)} \, \mathbf{\hat{y}} + \frac{c}{3}
\, \mathbf{\hat{z}},
\end{eqnarray}
and the volume of the primitive cell is one-third that of the conventional cell,
\begin{equation}
V = \left(\frac{2}{\sqrt3}\right) \, a^2 \, c.
\label{equ:vol11}
\end{equation}

The vectors above are all of identical length,
\begin{equation}
\label{equ:rhomba}
|\mathbf{a}_1| = |\mathbf{a}_2| = |\mathbf{a}_3| = \sqrt{\frac{a^2}{3} + \frac{c^2}{9}}
\equiv a',
\end{equation}
or, equivalently, $a = b = c \equiv a'$, where we designate the
common length as $a'$ to distinguish it from the length of the first
two vectors in the conventional lattice. The vectors also make
equal angles with each other
\begin{equation}
\label{equ:rhombg}
\alpha = \beta = \gamma = \cos^{-1} \left(\frac{2 c^2 - 3 a^2}{2 \left( c^2 + 3 a^2 \right)}\right).
\end{equation}
The above equations provide another
definition of the rhombohedral lattice. We can show this by writing
the primitive vectors in a form that depends only on the common length
and separation angle.
\begin{eqnarray}
\label{equ:lat11a}
\mathbf{a}_1 & = & a' \left(
\begin{array}{c}
\sin\frac{\alpha}{2} ~ \mathbf{\hat{x}} \\
- \left(\frac{1}{\sqrt{3}}\right) \, \sin\frac{\alpha}{2} ~ \mathbf{\hat{y}} \\
+ \sqrt{\frac{1}{3}\left(4 \cos^2\frac{\alpha}{2} - 1\right)} ~ \mathbf{\hat{z}}
\end{array}
\right) \nonumber \\
\mathbf{a}_2 & = & a' \left(
\begin{array}{c}
\left(2/\sqrt{3}\right) \, \sin\frac{\alpha}{2} ~ \mathbf{\hat{y}} \\
+ \sqrt{\frac{1}{3}\left(4 \cos^2\frac{\alpha}{2} - 1\right)} ~ \mathbf{\hat{z}}
\end{array}
\right) \nonumber \\
\mathbf{a}_3 & = & a' \left(
\begin{array}{c}
- \sin\frac{\alpha}{2} ~ \mathbf{\hat{x}} \\
- \left(\frac{1}{\sqrt{3}}\right) \, \sin\frac{\alpha}{2} ~ \mathbf{\hat{y}} \\
+ \sqrt{\frac{1}{3}\left(4 \cos^2\frac{\alpha}{2} - 1\right)} ~ \mathbf{\hat{z}}
\end{array}
\right).
\end{eqnarray}
An alternative orientation is given
by Setyawan and Curtarolo,
who only give the primitive vectors in this $(a',\alpha)$ setting. The primitive
vectors used for their rhombohedral cell differ
from the equations above only by the orientation of the vectors
relative to the Cartesian axes. Their choice is simpler for
computational purposes, but does not show the relationship between
the $(a,c)$ and $(a',\alpha)$ variants.

We can define the rhombohedral lattice in two ways: as a trigonal
lattice with additional translational vectors, or as a simple

lattice with equal primitive vectors making equal angles with one
another. The *International Tables* addresses this ambiguity by
listing atomic positions for the rhombohedral lattice in a
hexagonal setting,

where all coordinates are referenced to the
conventional cell, and in a rhombohedral
setting,

where the coordinates are referenced to
rhombohedral lattice. To further confuse matters, the unit cell's
dimensions might be reported in terms of $(a,c)$ or in terms of $(a',\alpha)$.
An article might say that there were N atoms in the
rhombohedral cell, or 3N atoms in the conventional cell. One has to
pay attention to the context.
In the database, we will report the lattice parameters of the system
by giving $a$ and $c$, since that is the usual crystallographic
practice. However, we will record atomic positions using the
rhombohedral primitive vectors, since computer calculations
work best with the smallest number of atoms needed to describe the
system.

The space groups associated with the rhombohedral lattice are \begin{array}{lll} 146. ~ \mbox{R3} & 148. ~ \mbox{R$\overline{3}$} & 155. ~ \mbox{R32} \\ 160. ~ \mbox{R3m} & 161. ~ \mbox{R3c} & 166. ~ \mbox{R$\overline{3}$m} \\ 167. ~ \mbox{R$\overline{3}$c} & ~ & ~ \\ \end{array}