Estoy tratando de inyectar mi UserDAO dentro de mi AuthorizingRealm personalizado que está utilizando Apache Shiro pero ... me sale nulo.

¿Qué estoy haciendo mal?

shiro.ini

[main]
user = demo.shiro.security.FacesAjaxAwareUserFilter
realmA = demo.shiro.security.JpaRealm
credentialsMatcher = org.apache.shiro.authc.credential.SimpleCredentialsMatcher
realmA.credentialsMatcher = $credentialsMatcher
securityManager.realms = $realmA
user.loginUrl = /pages/public/login.xhtml

[users]
admin = admin
user = user

[urls]
# public files and folders
/index.html = anon
/resources/** = anon
/pages/public/** = anon

# restricted files and folders
/pages/admin/** = user
/pages/user/** = user

JpaRealm.java

public class JpaRealm extends AuthorizingRealm {

    @Inject
    private UserDao userDao;

    public JpaRealm() {
        setCredentialsMatcher(new Sha256CredentialsMatcher());
    }

    protected AuthenticationInfo doGetAuthenticationInfo(AuthenticationToken authToken) throws AuthenticationException {
        UsernamePasswordToken token = (UsernamePasswordToken) authToken;
        User user = userDao.getForUsername(token.getUsername());
        if (user != null) {
            return new SimpleAuthenticationInfo(user.getId(), user.getPassword(), getName());
        } else {
            return null;
        }
    }

    protected AuthorizationInfo doGetAuthorizationInfo(PrincipalCollection principals) {
        Long userId = (Long) principals.fromRealm(getName()).iterator().next();
        User user = userDao.findByKey(userId);
        if (user != null) {
            SimpleAuthorizationInfo info = new SimpleAuthorizationInfo();
            for (Role role : user.getRoles()) {
                info.addRole(role.getDescription());
                for (Permition permition : role.getPermitions()) {
                    info.addStringPermission(permition.getDescription());
                }
            }
            return info;
        } else {
            return null;
        }
    }

}

¿Qué debo hacer para permitir que CDI tenga conocimiento del @Inject dentro de mi ámbito personalizado e inyectar mi UserDAO correctamente?