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Suppose that ${\\mathfrak q}\\colon \\mathcal{T}\\times \\mathcal{T}\\longrightarrow \\mathcal{T}$ is an $\\mathcal{R}$-bilinear mapping and that ${\\mathfrak T}_{\\mathfrak q}\\colon: \\mathcal{T}\\longrightarrow \\mathcal{T}$ is a trace of $\\mathfrak{q}$. We describe the form of ${\\mathfrak T}_{\\mathfrak q}$ satisfying the condition $[{\\mathfrak T}_{\\mathfrak q}(T), T]\\in \\mathcal{Z(T)}$ for all $T\\in \\mathcal{T}$. 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