  Let $\mathcal{H} \subset \mathcal{H}_{n,d} := \mathbb{R}[x_1$,$\ldots$,
$x_n]_d$ be a vector space, and $A$ be a compact semialgebraic subset of
$\mathbb{P}_{\mathbb{R}}^{n-1}$. We shall study some PSD cones $\mathcal{P} =
\mathcal{P}(A$, $\mathcal{H}) := \big\{f \in \mathcal{H}$ $\big|$ $f(a) \geq 0$
($\forall a \in A$)$\big\}$. Our interests are (1) to determine the extremal
elements of $\mathcal{P}$, (2) to determine discriminants of $\mathcal{P}$, (3)
to describe $\mathcal{P}$ as a union of basic semialgebraic subsets, and (4) to
find a nice test set when $\dim \mathcal{H}$ is low. In this article, we
present (1), (2), (3) and (4) for $\mathcal{P}(\mathbb{R}^4$,
$\mathcal{H}_{4,4}^{s0})$ and $\mathcal{P}(\mathbb{R}_+^4$,
$\mathcal{H}_{4,4}^{s0})$, where $\mathcal{H}_{n,d}^{s0} := \big\{f \in
\mathcal{H}_{n,d}$ $\big|$ $f$ is symmetric and $f(1,\ldots,1)=0 \big\}$. We
also provide (1) -- (4) for $\mathcal{P}(\mathbb{R}_+^4$,
$\mathcal{H}_{4,3}^{c0})$, where $\mathcal{H}_{n,d}^{c0} := \big\{f \in
\mathcal{H}_{n,d}$ $\big|$ $f$ is cyclic and $f(1,\ldots,1)=0 \big\}$.
