The real Diamond Lie algebra $\mathfrak{D}$ is a four-dimensional Lie algebra with basis $\left\{ J, P_{1}, P_{2}, T \right\}$ and non-zero relations $$ \left[J, P_{1}\right]=P_{2}, \quad\left[J, P_{2}\right]=-P_{1}, \quad\left[P_{1}, P_{2}\right]=T $$ The complexification of the Diamond Lie algebra: $\mathfrak{D} \otimes_{\mathbb{R}} \mathrm{C}$ displays the following complex basis: $$ \left\{P_{+}=P_{1}-i P_{2}, \quad P_{-}=P_{1}+i P_{2}, \quad T, \quad J\right\} $$ where $i$ is the imaginary unit, whose nonzero commutators are $$ \left[J, P_{+}\right]=i P_{+}, \quad\left[J, P_{-}\right]=-i P_{-}, \quad\left[P_{+}, P_{-}\right]=2 i T $$
An ideal $P$ of Lie algebra $L$ is called prime if $[H, K] \subseteq P$ with $H, K$ ideals of $L$ implies $H \subseteq P$ or $K \subseteq P$
How can I determine the ideals and compute the relations between ideals to know the prime ideals in $\mathfrak{D}$?
Are there a way to determine the ideals (In general in any Lie algebra not specifically $\mathfrak{D}$) or compute its relation by using the commutators of the basis as given above ?