We have collected in 2004 adaptive optics corrected L' images of the Galactic Center region with NAOS-CONICA at VLT. A strong variability was observed as well as a correlation between the photocenter positions and fluxes of the L'-band counterpart of Sgr A*. It is interpreted as the combined emission of a point-like flaring source at the position of Sgr A*/IR itself and an extended dust structure, 75 mas south west of Sgr A*/IR, which we name Sgr A*-f. We examine the different possible mechanisms to explain this dual Sgr A* L' emission and conclude it is likely a flaring emission associated to energetic events in the close environment of the black hole plus a quiescent emission resulting from the collision of Sgr A*-f by a jet from Sgr A*.