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Let 
$R$ be an 
$n!$ -torsion free semiprime ring with involution 
$*$ and with extended centroid 
$C$ , where 
$n\,>\,1$ is a positive integer. We characterize 
$a\,\in \,K$ , the Lie algebra of skew elements in $R$ , satisfying 
${{(\text{a}{{\text{d}}_{a}})}^{n}}\,=\,0$ on 
$K$ . This generalizes both Martindale and Miers’ theorem and the theorem of Brox et al. In order to prove it we first prove that if 
$a,\,b\,\in \,R$ satisfy 
${{(\text{a}{{\text{d}}_{a}})}^{n}}\,=\,\text{a}{{\text{d}}_{b}}$ on $R$ , where either 
$n$ is even or 
$b\,=\,0$ , then 
${{(a\,-\,\lambda )}^{[(n+1)/2]}}\,=\,0$ for some 
$\lambda \,\in \,C$ .
