Compound $\mathbf{A}$ is an optically active alcohol. Treatment with chromic acid converts $\mathbf{A}$ into a ketone, $\mathbf{B}$. In a separate reaction, $\mathrm{A}$ is treated with $\mathrm{PBr}_{3}$, converting A into compound C. Compound $\mathbf{C}$ is purified, and then it is allowed to react with magnesium in ether to give a Grignard reagent, D. Compound $\mathbf{B}$ is added to the resulting solution of the Grignard reagent. After hydrolysis of the initial product $(\mathbf{E})$, this solution is found to contain 3,4 -dimethylhexan- $3-$ ol. Propose structures for compounds $\mathbf{A}, \mathbf{B}, \mathbf{C}, \mathbf{D}$, and $\mathbf{E}$.