im confused on when should we include Beta asset (Ba) in the Ke formula. Most examples i see that the Ba is used to calculate the Be but are there instances whereby the Ba is directly included in the Ke?
The equity beta incorporates both business risk and the gearing risk, and it is always the equity beta that determines the return on equity (and hence the cost of equity). The asset beta measures the business risk and does not include the gearing risk.
The equity beta is only equal to the asset beta if there is no gearing. If there is no gearing, it is therefore the asset beta that determines the return on equity.
All of this is explained in my free lectures on CAPM.