  Let $P = \Bbbk[x_1, x_2, x_3]$ be a unimodular quadratic Poisson algebra,
with its Poisson bracket written as $\{x_i, x_j\} =
\displaystyle{\sum_{k,l}c_{i,j}^{k,l}x_kx_l}$, $1 \leq i < j \leq 3$. Let
$P_{\hbar}$ be the deformation quantization of $P$ constructed as follows:
$P_{\hbar} = \Bbbk\langle y_1, y_2,
y_3\rangle/([y_i,y_j]=\frac{\hbar}{2}\displaystyle{\sum_{k,l}}c_{i,j}^{k,l}(y_ky_l+y_ly_k))_{1
\leq i < j \leq 3}$. In this paper, we establish that $P$ and $P_{\hbar}$
possess identical graded automorphisms and reflections, and that taking
invariant subalgebras and taking deformation quantizations are two commutative
processes.
