We present 1.35 mm Submillimeter Array (SMA) observations around the low-mass Class 0 source IRAS 19156+1906, at the center of the LDN 723 (L723) dark cloud. We detected emission from dust as well as emission from H₂CO 3_0,3-2_0,2, DCN 3-2, and CN 2-1 lines, which arise from two cores, SMA 1 and SMA 2, separated by 29 (880 AU in projected distance). SMA 2 is associated with the previously detected source VLA 2. Weak SiO 5-4 emission is detected, possibly tracing a region of interaction between the dense envelope and the outflow. We modeled the dust and H₂CO emission from the two cores. The results from the modeling show that the cores have similar physical properties (density and temperature distribution) but that SMA 2 has a larger p-H₂CO abundance (by a factor of 3-10) than SMA 1. The p-H₂CO abundances' findings are compatible with the value of the outer part of the circumstellar envelopes associated with Class 0 sources. SMA 2 is harboring an active multiple low-mass protostellar system and powering at least one molecular outflow. In contrast, there are no known signs of outflow activity toward SMA 1. This suggests that SMA 2 is more evolved than SMA 1. The kinematics of the two sources show marginal evidence of infall and rotation motions. The mass detected by the SMA observation, which trace scales of 1000 AU, is only a small fraction of the mass contained in the large-scale molecular envelope, which suggests that L723 is still in a very early phase of star formation. Despite the apparent quiescent nature of the L723, fragmentation is occurring at the center of the cloud at different scales. Thus, at 1000 AU, the cloud has fragmented in two cores: SMA 1 and SMA 2. At the same time, at least one of these cores, SMA 2, has undergone additional fragmentation at scales of 150 AU, forming a multiple stellar system.