Deformation of the Heisenberg Algebra inside $gl(3,\mathbb{K})$

We study non-trivial deformations of the natural imbedding of the Lie algebra ${\text{fh}}_1$ of lower triangular matrices (the Heisenberg Lie algebra) into $gl(3,\mathbb{K})$, where $\mathbb{K}=\mathbb{R}$ or $\mathrm{\mid }mathbbC$. Our first result is the calculation of the first cohomology space $H^1({\text{fh}}_1;gl(3,\mathbb{K}))$. We prove that there are no obstructions for integrability of infinitesemal deformation and, furthermore, give an explicit formula for the most general deformation.