Inspired by the work of Le Bruyn and Smith in (Proc. Amer. Math. Soc. 118(3): 725–730, 1993) and the work of Shelton and Vancliff in (Comm. Alg. 30(5): 2535-2552, 2002), we analyze certain graded algebras related to the Lie algebra $\mathfrak {sl}(2,\Bbbk)$ using geometric techniques in the spirit of Artin, Tate and Van den Bergh. In particular, we discuss the point schemes and line schemes of certain quadratic quantum $\mathbb {P}^{3}$ s associated to the Lie superalgebra $\mathfrak {sl}(1|1)$ , to a quantized enveloping algebra, $\mathcal {U}_q(\mathfrak {sl}(2,\Bbbk))$ , of $\mathfrak {sl}(2,\Bbbk)$ , and to a color Lie algebra $\mathfrak {sl}_k(2,\Bbbk)$ , respectively. The geometry we consider identifies certain normal elements in the universal enveloping algebra of $\mathfrak {sl}(1|1)$ and in $\mathcal {U}_q(\mathfrak {sl}(2,\Bbbk))$ .