Co-Se compounds were found to exhibit Al storage through conversion reactions . In CoSe2 based electrodes, the Al-ions partially substitute Co in the crystal structure, meaning that Co2+-ions would in turn dissolve into the electrolyte. In bulk CoSe2, there is little control over this process which leads to degradation of the electrode structure and thus capacity loss over time. 
In addition to the examples provided here, there are numerous other positive electrode materials under research for Al-ion batteries, including some exotic ones that can't be categorized in the above manner. Also, the examples provided are not fully investigated and may not be the best preforming solutions for an Al-ion battery positive electrode. Al-ion batteries are still in active development.
The majority of research on Al-ion batteries has been focused on exploring electrolytes and positive electrodes and far less the possible negative electrodes. However, the widely used Al foil negative electrode has some considerable drawbacks. A metallic Al foil electrode is susceptible to dendrite growth and uneven corrosion which results in its eventual pulverization after repeated charge discharge cycles , limiting the lifetime of Al-ion battery devices.
One alternative is using an intercalation mechanism, instead of plating, at the negative electrode as well. An example would be MnO3 as negative electrode . Another approach includes using a highly corrosion resistant material, such as Cu foil, as the basis for Al plating and stripping, eliminating the pulverization danger . However, this does not address the issue caused by dendrite growth. There are more examples, like TiO2 nanopowder and carbon black mixture , and N-doped carbon nanorod array  as negative electrodes, but none of them are extensively studied yet.
It is also possible to protect the Al foil from pulverization and suppress dendrite growth by creating a protective oxide layer on the foil surface. The Al-ions have to deposit under the oxide layer which suppresses dendrite growth and promotes a more even stripping of the Al foil. While the oxide layer is separating the electrode and electrolyte, the electrolyte can still penetrate through defects in the oxide layer. This enables a longer operating time for the Al foil electrode.  However, it is also reported that these protective oxide layers are corroded over time in ionic liquid electrolytes .
A novel idea for a negative electrode is using liquid Ga. As a liquid, Ga provides a uniform dendrite free surface. The Al-ions go through alloying and dealloying at the Ga electrode during charge and discharge respectively. While this material can successfully eliminate dendrite growth and other electrode degradation, a battery using this electrode material would mostly be suited for stationary applications only due to the liquid nature of the electrode.  In addition, the battery would effectively be unusable below 30οC.
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Plett, Gregory L., Battery Management Systems, Volume 1 - Battery Modeling, Artech House, 2015. Electronic ISBN: 978-1-5231-1698-0. Available at:
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