On multivariate skewness and kurtosis

Let $X$ be a $d$-dimensional standardized random variable $(\mathbf{E}(X)=0,\operatorname{cov}(X)=1)$. Then for a multivariate analogue of skewness $s=\mathbf{E}(\|X\|^2X)$ and kurtosis $k=\mathbf{E}XX^TXX^T-(d+2)I$ we show that $\|s\|^2\le\operatorname{tr}k+2d$. For infinitly divisible distributions $\|s\|^2\le\operatorname{tr}k$.