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In real-world dynamical systems, technical limitations may prevent complete access to their dynamical variables. Such a lack of information may cause significant problems, especially when monitoring or controlling the dynamics of the system is required or when decisions need to be taken based on the dynamical state of the system. Cross-predicting the missing data is, therefore, of considerable interest. Here, we use a machine learning algorithm based on reservoir computing to perform cross-prediction of unknown variables of a chaotic dynamical laser system. In particular, we chose a realistic model of an optically injected single-mode semiconductor laser. While the intensity of the laser can often be acquired easily, measuring the phase of the electric field and the carriers in real time, although possible, requires a more demanding experimental scheme. We demonstrate that the dynamics of two of the three dynamical variables describing the state of the laser can be reconstructed accurately from the knowledge of only one variable, if our algorithm has been trained beforehand with all three variables for a limited period of time. We analyze the accuracy of the method depending on the parameters of the laser system and the reservoir. Finally, we test the robustness of the cross-prediction method when adding noise to the time series. The suggested reservoir computing state observer might be used in many applications, including reconstructing time series, recovering lost time series data and testing data encryption security in cryptography based on chaotic synchronization of lasers. [ABSTRACT FROM AUTHOR]