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Let $U$ be the two-parameter quantized enveloping algebra $U_{r,s}\left({\mathrm{𝔰𝔩}}_2\right)$ and $F\left(U\right)$ the locally finite subalgebra of $U$ under the adjoint action. The aim of this paper is to determine some ring-theoretical properties of $F\left(U\right)$ in the case when $r{s}^{-1}$ is not a root of unity. Then we describe the annihilator ideals of finite dimensional simple modules of $U$ by generators.