General linear hypothesis for a restricted regression model

An analyst fits a multiple linear regression model $Y_i=\beta_0+\beta_1X_{1i}+\cdots+\beta_4X_{4i}+\epsilon_i$, $i=1,2,\dots,n$, using the method of least squares. However, his coordinator insists that the intercept and two regressors $Z_1=X_1+X_3$ and $Z_2=X_2-X_4$ are enough to represent the model. Suppose the coordinator's claim can be tested in the form of a general linear hypothesis, viz., $H_0:L^T\beta=0$ against $H_A:H_0$ is not true, where $\beta^T=(\beta_0,\beta_1,\dots,\beta_4)$. Suppose we have $n$ observations on the response $Y$ and each regressor. Further, assume that the errors with or without restrictions are independent $N(0,\sigma^2)$ variables. Then, which of the following statements are true?