In Industry 4.0 (I4), the Industrial Internet of Things (I2oT) data streams are prone to significant data manipulation risks. The integration of Blockchains (BC) with I2oT may become a solution preventing from this problem. This paper provides a blockchain-agnostic Blockchain I2oT (BI2oT) architecture called BIIT that allows developing a broad range of BC applications fully integrating Internet of Things (IoT). The mechanisms introduced in BIIT aim at solutions that provide data reliability, limit the computational overhead, and enhance energy efficiency. BIIT is evaluated through real-world experimentation.