  In this paper, we study orthogonal representations of simple graphs $G$ in
$\mathbb{R}^d$ from an algebraic perspective in case $d = 2$. Orthogonal
representations of graphs, introduced by Lov\'asz, are maps from the vertex set
to $\mathbb{R}^d$ where non-adjacent vertices are sent to orthogonal vectors.
We exhibit algebraic properties of the ideal generated by the equations
expressing this condition and deduce geometric properties of the variety of
orthogonal embeddings for $d=2$ and $\mathbb{R}$ replaced by an arbitrary
field. In particular, we classify when the ideal is radical and provide a
reduced primary decomposition if $\sqrt{-1} \not\in K$. This leads to a
description of the variety of orthogonal embeddings as a union of varieties
defined by prime ideals. In particular, this applies to the motivating case $K
= \mathbb{R}$.
